Essentials 


Crystallography 


By 
EDWARD  HENRY  KRAUS,  Ph.D. 

\\ 

Junior  Professor  of  Mineralogy  in  the 
University  of  Michigan 


THE 

UNIVERSITY 

OF 


WITH  427   FIGURES 


GEORGE  WAHR,   Publisher 

Ann  Arbor,  Mich. 

1906 


COPYRIGHT,  1906 

by 
EDWARD  H.   KRAUS 


EIL!8    PUBLISHING    COMPANY,    PRINTEI 
BATTIE    CRfctK.     MICH. 


PREFACE. 

This    text    is   intended   for   beginners   and  aims  to  present  the 
essential  features  of   geometrical  crystallography  from  a  standpoint 


•icm 


ERRATA. 

Page  26,  last  line,  for  78°  31 '  44"  read  70°  31'  44". 

,   ra:ma:aOO~\  .    ra:ma:OOa1 

Page  35,  line  8,  for  +  I —        I  read  +  I         -j- 

Page  37,  line  6  from  the  bottom,  for  effected  read  affected. 

4O2  462 

Page  42,  last  line,  for  A  =  -f  -     —  /  read  A.  =  +        - /. 

2  4 

Page  45,  line  3,  for  planes  read  plane. 
Page  47,  line  14,  for  triangle  read  triangles. 


roi--«xKx-i.v,.ft.i.r--v*~ 


ngures  nave  oeen  muwn 

have  been  taken  from  various  sources. 

I  am  indebted  to  Mr.  W.  F.  Hunt,  Instructor  in  Mineralogy, 
also  to  Messrs.  I.  D.  Scott  and  C.  W.  Cook,  Assistants  in  Mineralogy 
in  the  University  of  Michigan,  for  aid  in  the  reading  of  the  proof. 

EDWARD    H.    KRAUS. 
Mineralogical  Laboratory,  University  of  Michigan, 

Ann  Arbor,  Mich.,  June,  1906. 


196447 


EIL!8    PUBLISHING    COMPANY,    PRINTERS 
BATTIE    CRttK.    MICH. 


PREFACE. 

This  text  is  intended  for  beginners  and  aims  to  present  the 
essential  features  of  geometrical  crystallography  from  a  standpoint 
which  combines  the  ideas  of  symmetry  with  those  of  holohedrism, 
hemihedrism,  etc.  All  forms  possible  in  the  thirty-two  classes  of 
symmetry  are  discussed  even  though  representatives  of  several  of  the 
classes  have  not  as  yet  been  observed  among  minerals  or  artificial 
salts.  In  each  system,  however,  the  important  classes  are  indicated 
so  that  the  book  may  readily  be  used  for  abridged  courses. 

No  attempt  has  been  made  to  discuss  the  measurement  and 
projection  of  crystals  inasmuch  as  they  involve  a  somewhat  compre- 
hensive knowledge  of  forms  and,  hence,  cannot  be  treated  adequately 
in  an  introductory  course.  For  a  similar  reason  very  little  reference 
has  been  made  to  the  various  structural  theories  of  crystals.  How- 
ever, the  bibliography  of  the  more  important  works,  which  have  been 
published  in  English  and  German  during  the  past  thirty  years,  will 
prove  useful  to  those  who  may  desire  to  pursue  the  study  of  crystal- 
lography further  than  the  scope  of  this  text  permits. 

Free  use  has  been  made  of  the  various  standard  texts  on  crystal- 
lography, but  especial  acknowledgments  are  due  von  Groth,  Linck, 
Bruhns,  Tschermak,  and  E.  S.  Dana.  A  very  large  majority  of  the 
figures  have  been  drawn  especially  for  this  text,  the  others,  however, 
have  been  taken  from  various  sources. 

I  am  indebted  to  Mr.  W.  F.  Hunt,  Instructor  in  Mineralogy, 
also  to  Messrs.  I.  D.  Scott  and  C.  W.  Cook,  Assistants  in  Mineralogy 
in  the  University  of  Michigan,  for  aid  in  the  reading  of  the  proof. 

EDWARD    H.    KRAUS. 
Mineralogical  Laboratory,  University  of  Michigan, 

Ann  Arbor,  Mich.,  June,  1906. 


196447 


[in] 


TABLE   OF   CONTENTS. 

INTRODUCTION. 

PAGE 

Crystals   and   Crystalline   Structure 1 

Amorphous    Structure 1 

Some   Properties  of  Crystallized  and  Amorphous   Substances 2 

Crystallography    2 

Constancy  of  Interfacial  Angles 2 

Crystal    Habit 4 

Crystallographic  Axes 4 

Crystal     Systems 5 

Parameters    and    Parametral    Ratio 5 

Fundamental     Forms 6 

Combinations 7 

Axial  Ratio 7 

Rationality  of  Coefficients 9 

Symbols 10 

Symbols  of  Naumann  and  Dana 11 

Miller's   System 12 

Elements  of  Symmetry 12 

Planes   of   Symmetry 12 

Axes  of  Symmetry 13 

Center  of  Symmetry 14 

Angular  Position  of  Faces 14 

Classes  of  Symmetry 14 

Holohedrism    15 

Hemihedrism 15 

Tetartohedrism , 15 

Ogdohedrism 15 

Correlated   Forms 15 

Hemimorphism 16 

CUBIC  SYSTEM. 

Crystallographic   Axes 17 

Classes  of  Symmetry 17 

Hexoctahedral  Class   (1) 17 

Hextetrahedral    Class    (2) 26 

Dyakisdodecahedral  Class    (3) 32 

Pentagonal  Icositetrahedral   Class    (4) 36 

Tetrahedral   Pentagonal   Dodecahedral   Class    (5) 38 

HEXAGONAL  SYSTEM. 

Crystallographic   Axes 43 

Classes  of  Symmetry 44 

[V] 


VI  TABLE   OF    CONTENTS. 

Dihexagonal  Bipyramidal  Class  (6) 44 

Dihexagonal  Pyramidal  Class   (7) 53 

Hexagonal    Hemihedrisms 55 

Ditrigonal  Bipyramidal  Class   (8) 56 

Ditrigonal  Scalenohedral  Class    (9) 60 

Hexagonal  Bipyramidal  Class   (10) 6"> 

Hexagonal  Trapezohedral  Class   (11) 68 

Ditrigonal  Pyramidal  Class    (12) 70 

Hexagonal  Pyramidal  Class  (13) 74 

Hexagonal    Tetartohedrisms 76 

Trigonal  Bipyramidal  Class    (14) ...  76 

Trigonal    Trapezohedral    Class     (15) 81 

Trigonal  Rhombohedral  Class   (16) 85 

Trigonal    Pyramidal    Class    (17) 89 

TETRAGONAL  SYSTEM. 

Crystallographic   Axes 92 

Classes    of    Symmetry 92 

Ditetragonal   Bipyramidal  Class    (18) 93 

Ditetragonal  Pyramidal  Class    (19) 99 

Tetragonal    Hemihedrisms 101 

Tetragonal   Scalenohedral  Class    (20) 102 

Tetragonal  Bipyramidal  Class   (21) 105 

Tetragonal  Trapezohedral  Class    (22) 108 

Tetragonal  Pyramidal  Class  (23) 110 

Tetragonal   Bisphenoidal  Class    (24) Ill 

ORTHORHOMBIC  SYSTEM. 

Crystallographic   Axes 115 

Classes    of    Symmetry 115 

Orthorhombic  Bipyramidal  Class    (25) 115 

Orthorhombic   Pyramidal   Class    (26) 121 

Orthorhombic  Bisphenoidal   Class    (27) 123 

MONOCUNIC  SYSTEM. 

Crystallographic   Axes 126 

Classes  of  Symmetry 126 

Prismatic    Class    (28) .    .  126 

Domatic  Class  (29) 132 

Sphenoidal   Class    (30) 135 

TRICLINIC  SYSTEM. 

Crystallographic   Axes 138 

Classes  of  Symmetry 138 

Pinacoidal    Class     (31) 138 

Asymmetric  Class  (32) 142 


TABLE    OF    CONTENTS.  VII 

COMPOUND  CRYSTALS. 

Parallel    Grouping 144 

Twin    Crystals 144 

Common  Twinning  Laws 

Cubic    System 145 

Hexagonal   System 146 

Tetragonal    System 147 

Orthorhombic    System 147 

Monoclinic   System 147 

Triclinic    System 148 

Repeated    Twinning 148 

Mimicry 149 

TABULAR  CLASSIFICATION  OF  THE  THIRTY-TWO    CLASSES 
OF  CRYSTALS. 

Cubic    System    150 

Hexagonal   System    151 

Tetragonal  System   153 

Orthorhombic   System    154 

Monoclinic    System 155 

Triclinic    System K6 

Index  ....  .   157 


BIBLIOGRAPHY. 

In  the  following  list  will  be  found  some  of  the  more  recent  works 
which  treat  crystallography  more  or  less  in  detail. 

Bauer,  M.,  Lehrbuch  der  Mineralogie,   2d  Ed.,   Stuttgart,  1904. 

Bauerman,  H.,    Text  Book  of  Systematic  Mineralogy,  London,  1881. 

Baumhauer,   H.,    Das  Reich  der  Krystalle,   Freiburg  in  Br. ,    1884. 
Die    neuere     Entwickelung    der    Krystallographie, 
Leipzig,  1905. 

Brezina,  A.,  Methodik  der  Krystallbestimmung,  Vienna,  1884. 

Bruhns,    W.,    Elemente  der  Krystallographie,   Liepzig  and  Vienna, 
1902. 

Brush-Penfield,    Manual  of    Determinative  Mineralogy,    New  York, 
1898. 

Dana,  E.  S.,  Text  Book  of  Mineralogy,  New  Edition,  New  York,  1898. 

Fock,  A.,   Introduction  to  Chemical  Crystallography,   translated  by 
W.  J.  Pope,  Oxford,  1895. 

Goldschmidt,  V.,  Krystallographische  Winkeltabellen,    Berlin,    1897. 
Index  der  Krystallformen  der  Mineralien,     3  vols., 

Berlin,  1886-1891. 

Anwendung  der  Linearprojection  zum  Berechnen  der 
Krystalle,  Berlin,  1887. 

Groth,  P.,  Physikalische  Krystallographie,   4th  Ed.,   Leipzig,    1905. 
Einleitung  in  die  Chemische  Krystallographie,  Leip- 
zig, 1904.      English  Translation  by  H.  Marshall, 
New  York,  1906. 

Heinrich,  F. ,  Lehrbuch  der  Krystallberechnung,  Stuttgart,  1886. 

Klein,  C.,  Einleitung  in  die  Krystallberechnung,  Stuttgart,  1876. 

Klockmann,  F.,  Lehrbuch  der  Mineralogie,  3d  Ed.,  Stuttgart,    1903. 

Lewis,  W.  J.,  Crystallography,  Cambridge,  1899. 

Liebisch,  T. ,  Geometrische  Krystallographie,  Leipzig,  1881. 

Physikalische  Krystallographie,  Leipzig,  1891. 
Grundriss  der  Physikalischen  Krystallographie,  Leip- 
zig,   1896. 

Linck,  G.,  Grundriss  der  Krystallographie,  Jena,  1896. 

[IX] 


X  BIBLIOGRAPHY. 

Miers,  H.  A.,  Mineralogy,  London  and  New  York,  1902. 

Moses  and  Parsons,  Mineralogy,  Crystallography  and  Blowpipe 
Analysis,  Revised  Edition,  New  York,  1904. 

Moses,  A.  J.,  Characters  of  Crystals,  New  York,  1899. 

Naumann-Zirkel,  Elemente  der  Mineralogie,  1 4th  Ed.,  Leipzig,  1901. 

Neis,  A.,  Allgemeine  Krystallbeschriebung  auf  Grund  einer  verein- 
fachten  Methode  der  Krystallzeichnung,  Stutt- 
gart, 1895. 

Rose,  G.,  Elemente  der  Krystallographie,  3d  Ed.,  Revised  by  A. 
Sadebeck,  Berlin,  1873. 

Sadebeck,  A.,  Angewandte  Krystallographie,  Berlin,  1876. 

Schonflies,  A.,  Krystallsysteme  and  Krystallstructur,  Leipzig,  1891. 

Sohncke,  L.,  Entwickelung  einer  Theorie  der  Krystallstructur,  Leip- 
zig, 1879. 

Story-Maskelyne,  N.,  Crystallography,  Oxford,  1895. 

Tschermak,  G.,  Lehrbuch  der  Mineralogie,  6th  Ed.,  Vienna,  1905. 

Viola,  C.,  Grundzuge  der  Kristallographie,  Leipzig,  1904. 

Voigt,  W.,  Die  Fundamentalen  Physikalischen  Eigenschaften  del 
Krystalle,  Leipzig,  1898. 

Von  Kobell,  F. ,  Lehrbuch  der  Mineralogie,  6th  Ed.,  Leipzig,  1899. 

Websky,  M.,  Anwendung  der  Linearprojection  zum  Berechnen  del 
Krystalle,  Berlin,  1887. 

Williams,  G.  H.,  Elements  of  Crystallography,  3d  Ed.,  New  York 
1890. 

Wiilfing,  E.  A. ,  Tabellarische  Uebersicht  der  Einfachen  Formen  del 
32  Krystallographischen  Symmetriegruppen, 
Stuttgart,  1895. 

Also,   S.  L.  Penfield,   Stereographic  Projection   and  its   Possibilities 
from  a  Graphic  Standpoint.     American  Journa 
of  Science,  1901,  XI,  1-24  and  115-144. 
On  the  Solution  of  Problems  in  Crystallography  b) 
Means  of  Graphical  Methods,  Ibid.,  1902,  XIV, 
249-284. 
On  Crystal  Drawing,  Ibid.,  1905,  XIX,    39-75. 


UNIVERSITY 

*  Li  FOR N^ 


INTRODUCTION.^ 


Crystals  and  Crystalline  Structure,  Nearly  all  homogeneous 
substances  possessing  a  definite  chemical  composition,  when  solidify- 
ing from  either  a  solution,  state  of  fusion  or  vapor,  attempt  to  crys- 
tallize, that  is,  to  assume  certain  characteristic  forms.  If  the  process 
of  solidification  is  slow  enough  and  uninterrupted,  regular  forms, 
bounded  by  plane  surfaces,  usually  result.  These  regular  polyhedral 
forms  are  termed  crystals.  If,  however,  the  solidification  is  so  rapid 
that  the  substance  cannot  assume  well  defined  forms,  bounded  by 
plane  surfaces,  the  solid  mass,  which  results,  is  said  to  be  crystalline. 
Crystalline  substances  consist  of  aggregations  of  crystals,  which  have 
been  hindered  in  their  development.  Their  outline  and  orientation 
are,  hence,  irrregular.  Figure  I  shows  a  well  developed  crystal  of 


\ 


Fig-  1.  Fig.  2. 

calcite  (CaCO3),  while  figure  21)  represents  a  cross-section  through  a 
crystalline  aggregate  of  the  same  substance.  Here,  no  definite  outline 
is  to  be  recognized  on  the  component  parts  of  the  mass. 

Amorphous  Structure.     Those  substances,  however,   which  do 
not  attempt  to  crystallize,  when  solidifying,  are  termed  amorphous. 

i)    After  Weinschenk.  [1] 


2  INTRODUCTION. 

They  are  without  form.  Opal  (SiO2.  x  H2O),  for  example,  never 
occurs  in  welj.  .defined  crystals  or  in  crystalline  masses. 

':   .Some  -"Pro'pBrties of  Crystallized  and  Amorphous  Substances. 

There;  are; 'tnany  interesting  features  to  be  noted  in  the  study  of 
crystallized  and  amorphous  substances.  The  attempt  at  crystalliza- 
tion and  the  assuming  of  regular  polyhedral  forms,  which  is  charac- 
teristic of  bodies  belonging  to  the  first  class,  are  only  some  of  the 
expressions  of  the  nature  of  such  substances.  The  various  physical 
properties,  hardness,  elasticity,  cohesion,  transmission  of  heat  and 
light,  usually  vary  in  such  crystallized  substances  with  the  direction 
according  to  fixed  laws.  This  is  not  the  case  with  amorphous  sub- 
stances. Hence,  we  may  define  a  crystal  as  a  solid  body,  which 
is  bounded  by  natural  -plane  surfaces  and  whose  form  is 
dependent  upon  its  physical  and  chemical  properties. 

Crystallography.  This  science  treats  of  the  various  properties 
of  crystals  and  crystallized  bodies.  It  may  be  subdivided  as  follows: 

1.  Geometrical   Crystallography. 

2.  Physical  Crystallography, 
j.    Chemical  Crystallography. 

Geometrical  crystallography,  as  the  term  implies,  describes  the 
various  forms  occurring  upon  crystals.  The  relationships  existing 
between  the  crystal  form  and  the  physical  and  chemical  properties 
of  crystals  are  the  subjects  of  discussion  of  the  second  and  third  sub- 
divisions of  this  science,  respectively.  Only  the  essentials  of  the 
first  subdivision — geometrical  crystallography  —  will  be  treated  in 
this  text. 

Constancy  of  Interfacial  Angles.  As  indicated  on  page  i, 
crystals  may,  in  general,  result  from  solidification  from  a  solution, 
state  of  fusion,  or  vapor.  Let  us  suppose  that  some  ammonium 
alum,  (NH4)2A12  (SO4)4.24H2O,  has  been  dissolved  in  water  and  the 
solution  allowed  to  evaporate  slowly.  As  the  alum  begins  to  crystal- 
lize, it  will  be  noticed  that  the  crystals  are,  for  the  most  part,  bounded 
by  eight  plane  surfaces.  If  these  surfaces  are  all  of  the  same  size,  that 
is,  have  been  equally  developed,  the  crystals  will  possess  an  outline 
as  represented  by  figure  3.  Such  a  form  is  termed  an  octahedron. 
The  octahedron  is  bounded  by  eight  equilateral  triangles.  The 
angles  between  any  two  adjoining  surfaces  or  faces,  as  they  are  often 


INTRODUCTION.  3 

called,  is  the  same,  namely,   109°  28^'.     On  most  of  the  crystals, 


Fig.  3. 


Fig  4. 


Fig.  5. 


Fig.  6. 


Fig.  7. 


Fig.  8. 


however,  it  will  be  seen  that  the  various  faces  have  been  developed 

unequally,    giving     rise  to  the    forms    illustrated   by   figures  4  and 

5.     Similar  cross-sections 

through  these   forms   are 

shown  in  figures  6,  7,  and 

8,   and  it  is  readily  seen 

that,    although    the    size 

of   the  faces  and,  hence, 

the  resulting  shapes  have 

been  materially  changed, 

the    angle    between    the 

adjoining    faces    has    re-  F-     9 

mained  the  same,  namely, 

109°  28^'.     Such  forms 

of     the    octahedron    are 

said  to  be  misshapen  or 

distorted.     Dist  or  t  ion 

is    quite   common    on  all 

crystals  regardless  of  their 

chemical  composition. 

It    was    the    Danish 
physician    and    natural 


Fig.  11. 


Fig.  12. 


INTRODUCTION. 


scientist,  Nicolas  Steno,  who  in  1669  first  showed  that  the  angles 
between  similar  faces  on  crystals  of  quartz  remain  constant  regard- 
less of  their  development.  Figures  9  and  10  represent  two  crystals 
of  quartz  with  similar  cross-sections  (figures  11  and  12).  Further 
observations,  however,  showed  that  this  applies  not  only  to  quartz  but 
to  all  crystallized  substances  and,  hence,  we  may  state  the  law  as 
follows:  Measured  at  the  same  temperature,  similar  angles  on 
crystals  of  the  same  substance  remain  constant  regardless  of 
the  size  or  shape  of  the  crystal. 

Crystal  Habit.  The  various  shapes  of  crystals,  resulting  from 
the  unequal  development  of  their  faces,  are  oftentimes  called  their 
habits.  Figures  3,  4,  and  5  show  some  of  the  habits  assumed  by 
alum  crystals.  In  figure  3,  the  eight  faces  are  about  equally 
developed  and  this  may  be  termed  the  octahedral  habit.  The  tab- 
ular  habit,  figure  4,  is  due  to  the  predominance  of  two  parallel 
faces.  Figure  5  shows  four  parallel  faces  predominating,  and  the 
resulting  form  is  the  prismatic  habit. 

Crystallographic  Axes.  Inasmuch  as  the  crystal  form  of  any 
substance  is  dependent  upon  its  physical  and  chemical  properties,  page 

2,  it  necessarily  follows  that  an  almost 
infinite  variety  of  forms  is  possible.  In 
order,  however,  to  study  these  forms  and 
define  the  position  of  the  faces  occurring  on 
them  advantageously,  straight  lines  of 
definite  lengths  are  assumed  to  pass  through 
the  ideal  center  of  each  crystal.  These  lines 
are  the  crystallographic  axes.  Their 
intersection  forms  the  axial  cross.  Figure 
13  shows  the  octahedron  referred  to  its 
three  crystal  axes.  In  this  case  the  axes 
are  of  equal  lengths  and  termed  a  axes. 
The  extremities  of  the  axes  are  differenti- 
ated by  the  use  of  the  plus  and  minus  signs, 
as  shown  in  figure  13. 

If  the  axes  are  of  unequal  lengths,  the 
one  extending  from  front  to  rear  is  termed 
the  a  axis,  the  one  from  right  to  left  the  by 


Fig.  13. 


+6.' 


-c 

Fig.  14. 


INTRODUCTION.  5 

while  the  vertical  axis  is  called  the  c  axis.  This  is  illustrated  by 
figure  14.  The  axes  are  always  referred  to  in  the  following  order, 
viz:  a,  b,  c. 

Crystal  Systems.  Although  a  great  variety  of  crystal  forms  is 
possible,  it  has  been  shown  in  many  ways  that  all  forms  may  be 
classified  into  six  large  groups,  called  crystal  systems.  In  the  group- 
ing of  crystal  forms  into  systems,  we  are  aided  by  the  crystallographic 
axes.  The  systems  may  be  differentiated  by  means  of  the  axes  as 
follows : 

1.  Cubic   System.     Three  axes,  all  of  equal  lengths,  intersect 
at  right  angles.     The  axes  are  designated  by  the  letters,  #,  a,  a. 

2.  Hexagonal   System.     Four  axes,  three  of  which  are  equal 
and  in  a  horizontal  plane  intersecting  at  angles  of  60°.       These  three 
axes  are  often  termed  the  lateral  or  secondary  axes,  and  are  desig- 
nated by  a,  a,  a.      Perpendicular  to  the  plane  of   the  lateral  axes  is 
the  vertical  axis,    which  may  be  longer  or  shorter  than  the  a  axes. 
This  fourth  axis  is  called  the  ^principal  or  c  axis. 

3.  Tetragonal  System.        Three  axes,  two  of  which  are  equal, 
horizontal,  and  perpendicular  to  each  other.      The  vertical,  c,  axis  is 
at  right  angles  to  and  either  longer  or  shorter  than  the  horizontal  or 
lateral,  a,  axes. 

4.  Orthorhombic   System,       Three    axes  of   unequal    lengths 
intersect  at  right  angles.     These  axes  are   designated  by  a,  b,  c,   as 
shown  in  figure  14. 

5.  Monoclinic  System.       Three    axes,    all   unequal,    two    of 
which  (#',  c)  intersect   at  an  oblique  angle,   the  third   axis  (b)  being 
perpendicular  to  these  two. 

6.  Triclinic   System,      Three   axes  (a,  b,    c)  are   all  unequal 
and  intersect  at  oblique  angles. 

Parameters  and  Parametral  Ratio,  In  order  to  determine  the 
position  of  a  face  on  a  crystal,  it  must  be  referred  to  the  crystal- 
lographic axes.  Figure  I  5  shows  an  axial  cross  of  the  orthorhombic 
system.  The  axes,  a,  b,  c,  are,  therefore,  unequal  and  perpendic- 
ular to  each  other.  The  plane  ABC  cuts  the  three  axes  at  the  points 
A,  B,  and  C,  hence,  at  the  distance  OA— a,  OB=b,  OC=c,  from 
the  center,  O.  These  distances,  OA,  OB,  and  OC,  are  known  as 


6 


INTRODUCTION. 


the  parameters  and  the  ratio,  OA  :  OB  :  OC,  as  the  parametral 
ratio  of  the  plane  ABC.  This  ratio  may,  however,  be  abbreviated 
to  a  :  b  :  c. 


Fig.  15. 


Fig.  16. 


There  are,  however,  seven  other  planes  possible  about  this  axial 
cross  which  possess  parameters  of  the  same  lengths  as  those  of  the 
plane  ABC,  figure  16.  The  simplified  ratios  of  these  planes  are: 


a 

-b 

c 

a 

b 

-c 

a 

-b 

-c 

-a 

b 

c 

-a 

-b 

c 

-a 

b 

-c 

-a 

-b 

-c 

These  eight  planes,  which  are  similarly  located  with  respect  to  the 
crystallographic  axes,  constitute  a  crystal 
form,  and  may  be  represented  by  the  general 
ratio  (a  :  b  :  c).  The  number  of  faces  in  a 
crystal  form  depends,  moreover,  not  only  upon 
the  intercepts  or  parameters  but  also  upon 
the  elements  of  symmetry  possessed  by  the 
crystal,  see  page  12.  Those  forms,  which 
enclose  space,  are  called  closed  forms. 
Figure  16  is  such  a  form.  Those,  however, 
which  do  not  enclose  space  on  all  sides,  as 
Fig.  17.  shown  in  figure  17,  are  termed  open  forms. 

Fundamental  Form.     In  figure  18,  the  enclosed  form  possesses 
the  general    ratio,  a  :  b  :  c.     The    face    ABM,    however,    has   the 


INTRODUCTION. 


parametral  ratio,  oA  :  oB  :  oM, 
where  oA  =  «,  oB  =  £,  and  oM  = 
3oC  — 3<7.  Hence,  this  ratio  may 
be  written  a  \  b  \  $c.  But,  as  in 
the  previous  case,  this  ratio  repre- 
sents a  form  consisting  of  eight  faces 
as  shown  in  the  figure.  That  form, 
the  parameters  of  which  are  selected 
as  the  unit  lengths  of  the  crystallo- 
graphic  axes,  is  known  as  the  unit 
or  fundamental  form.  In  figure 
1 8  the  inner  pyramid  is  a  so-called 
unit,  whereas  the  other  one  is  a 
modified  pyramid. 

Combinations.    Several  differ- 
ent forms  may  occur  simultaneously 
upon  a  crystal,  giving  rise  to  a  com- 
bination.    Figure  19  shows  a  com- 
bination of  two  pyramids  observed 
on  sulphur;    p  = 
a  :  b  :  c  (unit)  and 
s  =  a  :    b  :    Y^c 
(modified).  Figure 
20  shows  the  two 
forms,  o  =  a:a:a, 
and  h  =  a:    oca 
:  ooa,  seepage  19. 

Axial  Ratio. 

If   the    intercepts 

of    a    unit      form 

cutting   all  three  axes   be  expressed  in  figures,    the  intercept  along 

the  b  axis  being  considered  as  unity,  we  obtain  the  axial  ratio.     In 

figure  19,  which  represents  a  crystal  of  sulphur,  the  axial  ratio  is: 

a  :  b  :  c=  .8131  :  I  :  1.9034-" 

Every  crystallized  substance  has  its  own  axial  ratio.  This  is 
illustrated  by  the  ratios  of  three  minerals  crystallizing  in  the  ortho- 
rhombic  system. 


Fig.  19. 


Fig.  18. 


Fig.  20. 


8 


INTRODUCTION. 


Aragonite,  CaCO3,  a  :  b  :  c  =  .6228  :  i  :     .7204 

Anglesite,   PbSO4,  a  :  b  :  c  =  .7852  :  I  :  1.2894 

Topaz,  A12(F.OH)2  SiO4 ,  a  :  b  :  ^  =  .5281  :  I  :     .9442 

In  the  hexagonal  and  tetragonal  systems,  since  the  horizontal 
axes  are  all  equal,  i.  c.,  a  =  b,  see  page  5,  the  axial  ratio  is  reduced 
to  a  :  c;  a  now  being  unity.  Thus,  the  axial  ratio  of  zircon  (ZrSiO4) 
which  is  tetragonal,  may  be  expressed  as  follows:  a  :  c=  i  :  .6404; 
that  of  quartz  (SiO2),  hexagonal,  by  a  :  c  =  I  :  1.0999.  Obviously, 
in  the  cubic  system,  page  5,  where  all  three  axes  are  equal,  this  is 
unnecessary. 

However,  in  the  monoclinic  and  triclinic  systems,  where  either 
one  or  more  axes  intersect  obliquely,  it  is  not  only  necessary  to  give 
the  axial  ratio  but  also  to  indicate  the  values  of  the  angles 
between  the  crystallographic  axes.  For  example,  gypsum  (CaSO4. 
2H2O)  crystallizes  in  the  monoclinic  system  and  has  the  following 
axial  ratio: 

a'  :  b  :  c  —  .6896  :  I  :  .4133 

and  the  inclination  of  the  a'  axis  to  the  c  is  98°  58'.       This  angle 
is  known  as  £,  figure  2 1 . 


-b- 


-c 

Fig.  21, 


•fb 


Fig.  22. 


In  the  triclinic  system,  since  all  axes  are  inclined  to  each  other, 
it  is  also  necessary  to  know  the  value  of  the  three  angles,  which  are 
located  as  shown  in  figure  22,  viz:  b/\c=a,  a  A  c  =  /?,  a  A  b  =:y. 
The  axial  ratio  and  the  angles  showing  the  inclination  of  the  axes  are 
termed  the  elements  of  crystallization. 

The  triclinic  mineral  albite  (NaAlSi3O8)  possesses  the  following 
elements  of  crystallization: 

a:5:t=  .6330  :  I  :  5573. 

«=    94°     5' 
0=  116°  27' 

y=    88°     7' 


INTRODUCTION. 


9 


If  the  angles  between  the  crystallographic  axes  equal  90°,  they 
are  not  indicated.  Therefore,  in  the  tetragonal,  hexagonal,  and 
orthorhombic  systems,  the  axial  ratios  alone  constitute  the  elements 
of  crystallization,  while  in  the  cubic  system,  there  are  no  unknown 
elements. 


Fig.  23. 

Rationality  of  Coefficients.  The  parametral  ratio  of  any  face 
may  be  expressed  in  general  by  na  :  pb  :  me,  where  the  coefficients 
n,  p,  m<  are  according  to  observation  always  rational.  In  figure  23, 
the  inner  pyramid  is  assumed  to  be  the  fundamental  form, 
page  7,  with  the  following  value  of  the  intercepts:  oa=i.256, 
ob  =  i,  oc  —  .752.  But  the  coefficients  n,  p,  m,  are  obviously  all 
equal  to  unity.  The  ratio  is,  Hence,  a  :  b  :  c. 

The  outer  pyramid,  however,  possesses  the  intercepts,  oa  = 
1.256,  oJ3  =  2,  oC=2.2$6.  These  lengths,  divided  by  the  unit 


10 


INTRODUCTION. 


lengths  of  each  axis,  as  indicated  above,   determine  the  values  of  n, 
p,  and  m  for  the  outer  pyramid,  namely: 


=  3. 

1.256  i  .752 

These  values  of  n,  p,  and  ;w,  are,  therefore,  rational.  Such 
values  as  £,_J,  £,  f,  or  f  are  also  possible,  but  never  3.1416-!-, 
2.6578-)-,  1/3,  and  so  forth. 

Symbols.  The  parametral  ratio  of  the  plane  ABM,  figure  18, 
may  be  written  as  follows: 

na  :pb  :  me. 

But  since,  in  this  case,  n  =  i,  p  =  i,  m  =  3,  the  ratio  becomes: 

a  :  I  :  fa 

If,  however,  the  coefficients  had  the  values  |,  f  ,  and  J,  respect- 
ively, the  ratio  would  then  read: 

\a  :  \b  :  fa 

This,  when  expressed  in  terms  of  b,  becomes: 

\a  :  b  :  2c. 
Hence,  the  ratio 

na  :  b  :  me 

expresses  the  most  general  ratio  or  symbol  for  forms  belonging  to  the 
orthorhombic,  monoclinic,  and  triclinic  systems.  In  the  hexagonal 
and  tetragonal  systems,  since  the  a  and  b  axes  are  equal,  this  general 
symbol  becomes, 

a  :  na  :  me. 


Fig.  24. 


Figure  24  shows 
a  form,  the  ditetra- 
g  o  n  a  1  bipyramid, 
with  the  symbol  a  : 
2a  :  \c.  In  the  cubic 
system,  all  three  axes 
are  equal  and  the 
general  symbol 
reads, 

a  :  na  :  ma. 


INTRODUCTION. 


11 


I 

^^^ 

1 

1 
~!  
1 
1 
1 
1 

1 
1 

=K 

1 

i 
i 

— 

The  ratio  a  :  00  a  :  <X>a,  for  example,  symbolizes  a  form  in  the 
cubic  system  consisting  of  six  faces,  which  cut  one  axis  and  extend 
parallel  to  the  other  two.  Such  a  form 
is  the  cube,  figure  25.  The  ratio  a:2a:  CD  a 
represents  a  form  with  twenty-four  faces; 
each  face  cuts  one  axis  at  a  unit 's  length, 
the  second  at  twice  that  length,  but 
extends  parallel  to  the  third  axis.  Figure 
26  shows  such  a  form,  the  tetrahexahe- 
dron.  This  system  of  crystallographic 
notation  is  known  as  the  Weiss  system 
after  the  inventor  Prof.  C.  S.  Weiss. 
These  symbols  are  most  readily  under- 
stood and  well  adapted  for  beginners. 

Naumann,  Dana,  and  Miller  have  intro- 
duced modifications  tending  to  shorten  the 
symbols  of  Weiss.  Since  these  shortened 
forms  are  employed  quite  extensively,  it  will 
be  necessary  to  explain  each  briefly. 

Symbols  of  Naumann    and   Dana.     In 

the    notation    of    Naumann,    O    and    P,    the 

initial  letters  of  the  words,  octahedron  and  pyramid,  respectively, 
are  used  as  a  basis.  O  is  used  for  forms  of  the  cubic  system,  P  for 
all  other  systems.  The  coefficient  m,  referring  to  the  vertical  axis, 
is  placed  before  and  the  other  coefficient  n,  after  one  of  these  letters. 
For  example,  na  :  TJ  :  m£,  becomes  mPn,  and  a  :  ma  :  ooa,  CQQm. 

Dana 's  notation  is  similar  to  that  of  Naumanrr.  Dana,  however, 
substitutes  a  short  dash  for  the  letters  O  and  P,  also  i  or  I  for  oo, 
the  sign  of  infinity.  Otherwise,  the  two  systems  are  alike. 

The  following  table  shows  several  Weiss  symbols  with  the  cor- 
responding Naumann  and  Dana  modifications: 


Fig.  26. 


Weiss 


a 

b 

a 

I 

a 

2a 

a 

oca 

a 

2a 

a 

na 

^ 

2C 
\C 

oca 
oca 
ma 


Naumann 

2Pf 

|Pa 

ocOoo 

OCO2 

mOn 


Dana 
3 


- 

i-2 
m  —  n 


12 


INTRODUCTION. 


Miller's  System.  In  this  system,  as  is  also  the  case  with  the 
notations  of  Naumann  and  Dana,  th$  letters  referring  to  the  various 
crystallographic  axes  are  not  indicated.  The  values  given  being 
understood  as  referring  to  the  a,  b,  and  c  axes  respectively,  page  5. 
The  reciprocals  of  the  Weiss  parameters  are  reduced  to  the  lowest 
common  denominator,  the  numerators  then  constitute  the  Miller 
symbols,  called  indices^.  For  example,  the  reciprocals  of  the  Weiss 
parameters  2a  :  b  :  ^c  would  be  |,  y,  i-  These,  reduced  to  the  low- 
est common  denominator,  are  f ,  f,  f .  Hence,  362  constitute  the 
corresponding  Miller  indices.  These  are  read  three,  six,  two. 

A  number  of  examples  will  make  this  system  of  notation  clear. 
Thus,  a  :  dob  :  ccc,  becomes  100;  2a  :  b  :  $c,  5.10.2;  a  :  a  :  ?>c, 
331 ;  a  :  ooa  :  2c,  201,  and  so  forth.  The  Miller  indices  correspond- 
ing to  the  general  ratios  a  :  no,  :  ma  and  na  :  b  :  me  are  written 
hkl.  The  Miller  indices  are  important  because  of  their  almost 
universal  application  in  crystallographic  investigations.  The  trans- 
formation of  the  Naumann  or  Dana  symbols  to  those  of  Miller  should, 
after  the  foregoing  explanations,  present  no  difficulty  whatever. 

Elements  of  Symmetry.  The  laws  of. symmetry  find  expres- 
sion upon  a  crystal  in  the  distribution  of  similar  angles  and  faces. 
The  presence,  therefore,  of  planes,  axes,  or  a  center  of  symmetry  — 
these  are  the  elements  of  symmetry  —  is  of  great  importance  for  the 
correct  classification  of  a  crystal. 

Planes  of  Symmetry.  Any  plane,  which  passes  through  the 
center  of  a  crystal  and  divides  it  into  two  symme- 
trical parts,  the  one-half  being  the  mirror-image 
of  the  other,  is  a  plane  of  symmetry.  Figure  27 
shows  a  crystal  of  gypsum  (CaSO4.2H2O)  with 
its  one  plane  of  symmetry.  Every  plane  of  sym- 
metry is  parallel  to  some  face,  which  is  either 
present  or  possible  upon  the  crystal.  In  figure 
27  it  is  parallel  to  the  face  b. 

It  is  sometimes  convenient  to  subdivide  the 
planes  of  symmetry  into  principal  and  secondary 
or  common  planes,  according  to  whether  they 
possess  two  or  more  equivalent  and  interchangeable 
directions  or  not.  Figure  28  illustrates  a  crystal 
of  the  tetragonal  system  with  five  planes  of  sym- 


Fig.  27. 


INTRODUCTION. 


13 


metry.  The  horizontal  plane  c  is  the 
principal,  the  vertical  planes  the  second- 
ary planes  of  symmetry. 

Axes  of  Symmetry.  The  line,  about 
which  a  crystal  may  be  revolved  as  an  axis 
so  that  after  a  definite  angular  revolution 
the  crystal  assumes  exactly  the  same  posi- 
tion in  space  which  it  originally  had,  is 
termed  an  axis  of  symmetry.  Depending 
upon  the  rotation  necessary,  only  four  types 
of  axes  of  symmetry  are  from  the  stand- 
point of  crystallography  possible.1' 

a)  Those  axes,  about  which  the  original  position  is  reassumed 
after  a  revolution  of  60°,  are  said  to  be  axes  of  hexagonal,  six-fold, 
or  six-count2'*  symmetry.      Such  axes  may  be  indicated  by  the  sym- 
bol •.     Figure  29  shows  such  an  axis. 

b)  If  the  original  position  is  regained  after  the  crystal  is  revolved 
through  90°,  the  axis  is  termed  a  tetragonal,  four-count,  or  four- 
fold axis  of  symmetry.     These  axes  are  represented  by  •,  as  illus- 
trated in  figure  30. 


Fig.  28. 


*s 

!   T* 

1 

1 

i 

1 

I 

i 

1 

j 

i 

-k 

Fig.  29. 


Fig.  30. 


Fig.  31. 


c)  Axes  requiring  an  angular  revolution  of  120°  are  trigonal, 
three-fold,  or  three-count  axes  of  symmetry  and  may  be  symbolized 
by  A.  Figure  31  illustrates  this  type  of  axis. 


1)  For  proof,  see  Groth's  Physikalische  Krystallographie,  4te  Auflage,  1905,  321;  also  Viola'* 
Grundziige  der  Kristallographie,  1904,  251. 

2)  Because  in  a  complete  revolution  of  360°  the  position  is  reassumed  six  times. 


14 


INTRODUCTION. 


Fig  3*. 


d)  A  binary,  two-fold,  or  two-count  axis  necessitates  a  revo- 
lution through  1 80°.  These  are  indicated  by  •  in  figure  30. 

Center  of  Symmetry.  That  point  within  a 
crystal  through  which  straight  lines  may  be  drawn, 
so  that  on  either  side  of  and  at  the  same  distance 
from  it,  similar  portions  of  the  -crystal  (faces, 
edges,  angles,  and  so  forth)  are  encountered,  is  a 
center  of  symmetry.  Figure  32  has  a  center  of 
symmetry  —  the  other  elements  of  symmetry  are 
lacking. 

Angular  Position  of  Faces.  Since  crystals  are  oftentimes  mis- 
shapen or  distorted,  page  3,  it  follows  that  the  elements  of 

symmetry  are  not 
always  readily  recog- 
nized. The  angular 
•position  of  the  faces 
in  respect  to  these 
elements  is  the 
essential  feature,  and 
not  their  distance  or 
relative  size.  Fig- 
ure 33  shows  an  ideal 
crystal  of  a  u  g  i  t  e. 
Here,  the  presence 
Fig.  33.  Fig.  34.  of  a  plane  of  symme- 

try,   an  axis,   and  a 

center  of  symmetry  is  obvious.  Figure  34  shows  a  distorted  crystal  of 
the  same  mineral,  possessing  however  exactly  the  same  elements  of 
symmetry,  because  the  angular  position  of  the  faces  is  the  same  as  in 
figure  33. 

Classes  of  Symmetry.  Depending  upon  the  elements  of  sym- 
metry present,  crystals  may  be  divided  into  thirty-tivo  distinct  groups, 
called  classes  of  symmetry. l)  Only  forms  which  belong  to  the  same 
class  can  occur  in  combination  with  each  other.  A  crystal  system, 
however,  includes  all  those  classes  of  symmetry  which  can  be 
referred  to  the  same  type  of  crystallographic  axes,  page  5.  The 


l)    Also  termed  classes  of  crystals. 


INTRODUCTION. 


15 


various  elements  of  symmetry  and,  wherever  possible,  an  important 
representative  are  given  for  each  of  the  thirty-two  classes  in  the 
tabular  classification  on  page  150. 

Holohedrism.     Figure   35  shows  a  form  of   the  cubic  system. 
The  face    a  b  c    has  the  parametral  ratio  a  :  \a  :  30.      Forty-seven 
other  faces,  having  this  same  ratio,  are  how- 
ever also  possible  and,  when  present,  give  rise 
to  the  hexoctahedron,  as  the  complete  form 
is   called.     These    forty-eight    faces    are    all 
equal,  scalene  triangles.     Such  forms,  which 
possess  all  the  faces  possible  having  the  same 
ratio,  are  called  holohedral  forms. 


Fig.  35. 


Fiar  36. 


Hemihedrism.  If,  however,  one-half  of 
the  faces  of  the  hexoctahedron  be  suppressed 
and  the  other  half  be  allowed  to  expand,  as 
shown  in  figure  36,  the  diploid  with  half  of 
the  faces  possessed  by  the  hexoctahedron, 
namely  twenty-four,  results.  Forms  of  this 
character  are  said  to  be  hemihedral. 

Tetartohedrism.  Again,  if  only  one 
quarter  of  the  faces  expand,  all  others  being 
suppressed,  the  hexoctahedron  then  yields  the 
tetrahedral  pentagonal  dodecahedron  with 
but  twelve  faces,  figure  37.  This  is  a  tetarto- 
hedral  form. 

Ogdohedrism.  There  are  still  other 
forms,  which  possess  but  one-eighth  of  the 
number  of  faces  of  the  original  holohedral 
form,  and  these  are  termed  og-dohedral  forms. 
Figure  270,  page  91,  shows  such  ogdohedral 
forms  in  combination. 

It  is  evident  that  these  complete  and  partial  forms  possess  differ- 
ent elements  of  symmetry  and,  hence,  only  those  forms  which  are  of 
the  same  type,  that  is,  possess  the  same  elements  of  symmetry,  can 
enter  into  combination  with  each  other.  Compare  page  14. 

Correlated  Forms.  Obviously,  by  the  application  of  hemihe- 
drism,  tetartohedrism,  and  Ogdohedrism  the  original  holohedral  forms 


Fig.  37. 


16 


INTRODUCTION. 


yield  two,  four,  or  eight  new  correlated  forms,  respectively.  Most 
of  these  correlated  forms  are  congruent  and  differ  from  each  other 
only  in  respect  to  their  position  in  space.  A  rotation  through  some 
definite  angle  is  all  that  is  necessary  to  have  such  forms  occupy  the 
same  position.  Such  forms  are  then  designated  as  //^/s«and  minus, 
or  positive  and  negative,  forms,  page  27.  Others,  however,  are 
related  to  one  another  as  is  the  right  hand  to  the  left,  hence  cannot 
be  superimposed  and  these  are  said  to  be  enantiomorphous.  These 
are  designated  as  right  and  left  forms.  Compare  page  37. 


M 

Fig.  38.  Fig.  39.  Fig.  40. 

Hemimorphism.  This  is  a  peculiar  type  of  hemihedrism.  It  is 
only  possible  upon  those  holohedral  and  hemihedral  forms,  or  their 
combinations,  which  possess  a  so-called  singular**  axis  of  symmetry. 
By  its  application  the  forms  occurring  about  one  end  of  this  axis  differ 
from  those  about  the  other.  Such  forms  are  hemimorphic,  and  the 
singular  axis  of  symmetry  is  said  to  be  polar.  Hence,  the  plane  of 
symmetry  perpendicular  to  the  singular  axis  is  lost.  Compare  figures 
38,  39,  and  40. 


!)    An  axis  which  occurs    but    once    and    differs  from  all  others.    For  example,  the  axis  of 
tetragonal  symmetry  in  figure  30. 


OF    THE 


1 


CUBIC  SYSTEM" 

Crystallographic  Axes.  All  forms  which  can  be  referred  to 
three  equal  and  perpendicular  axes  belong  to  this  system.  Figure  41 

shows  the  axial   cross.      One    axis    is    held 

A.  A. 

vertically,  a  second  extends  from  front  to 
rear,  and  the  third  from  right  to  left. 
These  axes  are  all  interchangeable,  each 

± «.«      being  designated  by  a.     Since  there  are  no 

unknown  elements  of  crystallization  in  this 
system   (page   9),     all    substances,    regard- 
less of  their  chemical  composition,  crystal- 
p7*  41  lizing  in  this  system  with  forms  having  the 

same  parametral  ratios   must    of   necessity 
possess  the  same  interfacial  angles. 

Classes  of  Symmetry.  The  cubic  system  includes  five  groups 
or  classes  of  symmetry.  Beginning  with  the  class  of  highest  sym- 
metry, they  are: 

1)  Hexoctahedral  Class  (Holohedrism) 

2)  Hextetrahedral  Class  \ 

3)  Dyakisdodecahedral  Class  >•     (Hemihedrisni) 

4)  Pentagonal  icositetrahedral  Class     ) 

5)  Tetrahedral  pentagonal  dodecahedral  Class 

( Tetartohedrisrti) 

Of  these  classes,  the  first  three  are  most  important,  since  they 
possess  many  representatives  among  the  minerals. 

/.     HEXOCTAHEDRAL   CLASS.*)  }\  0  r  tt\  o\\  v 
(Holohedrism.} 

Elements  of  Symmetry,  a)  Planes.  Forms  of  this  class  are 
characterized  by  nine  planes  of  symmetry.  Three  of  these  are  par- 
allel to  the  planes  of  the  crystallographic  axes  and,  hence,  perpendic- 
ular to  each  other.  They  are  the  principal  planes  of  symmetry. 
They  divide  space  into  eight  equal  parts  called  octants.  The  six 


1)  Also  termed  the  regular,  isometric,  tesseral,  or  tessular  system. 

2)  Termed  by  Dana  the  normal  group. 


18 


CUBIC   SYSTEM. 


other  planes  are  each  parallel  to  one  of  the  crystallographic  axes  and 
bisect  the  angles  between  the  other  two.  These  are  termed  the  sec- 
ondary or  common  planes  of  symmetry.  By  them  space  is  divided 


i    i 


Fig.  42. 


Fig.  43. 


into  twenty-four  equal  parts.  The  nine  planes  together  divide  space 
into  forty-eight  equal  sections.  These  nine  planes  are  often  indicated 
thus:  3  -4-  6  =  9.  Figures  42  and  43  illustrate  the  location  of  the 
principal  and  secondary  planes,  respectively. 

b)  Axes.  The  intersection  lines  of  the  three  principal  planes 
of  symmetry  give  rise  to  the  three  principal  axes  of  symmetry. 
These  are  parallel  to  the  crystallographic  axes  and  possess  tetragonal 
symmetry,  as  illustrated  by  figure  44.  The  four  axes  equally  inclined 


<t^^ 

^?* 

r^           \                ? 

S 

\           / 

x  \     // 

//     ^  Xx 

^  /             N 

^ 

'  ^L-              V 

•-    ^ 

U'-'               ^ 

^ 

Fig.  44. 


Fig.  45. 


Fig.  46. 


to  the  crystallographic  axes  are  of  trigonal  symmetry,  as  shown  by 
figure  45.  There  are  also  six  axes  of  binary  symmetry.  These  lie 
in  the  principal  planes  of  symmetry  and  bisect  the  angles  ^  bet  ween 
the  crystallographic  axes.  Their  location  is  indicated  in  figure  46. 
These  thirteen  axes  of  symmetry  may  be  indicated  as  follows: 


c)     Center.      The  forms  of  this  class  also  possess  this  element  of 
symmetry.     Hence,  all  planes  have  parallel  counter-planes. 


Fig.  47. 


. 
HEXOCTAHEDRAL   CLASS.  19 

The  projection  of  the  most  general  form 
of  this  class  upon  a  plane  perpendicular  to 
the  vertical  axis,  i.  e.,  in  this  case  a  principal 
plane  of  symmetry,  shows  the  symmetry 
relations, 1}  figure  47. 

FORMS. 

1.  Octahedron,      As  the  name  implies, 
this  form  consists  of  eight  faces.     Each  face 
is  equally  inclined  to  the  crystallographic  axes. 
Hence,    the  parametral  ratio  may  be  written 
(a  :  a  :  a),   which  according  to  Naumann  and 
Miller  would  be  O  and   ji  i  i|,   respectively. 
The  faces  intersect  at  an  angle  of    109°  28' 
1 6"  and  in  the  ideal  form,  figure  48,  are  equal, 
equilateral  triangles. 

The  crystallographic  axes  and,  hence, 
the  axes  of  tetragonal  symmetry  pass  through 
the  tetrahedral  angles.  The  four  trigonal 
axes  join  the  centers  of  opposite  faces,  while 
the  six  binary  axes  bisect  the  twelve  edges. 

2.  Dodecahedron.     This  form   consists 
of    twelve    faces,    each    cutting    two    of    the 
crystallographic    axes    at    the    same  distance, 
but    extending    parallel    to    the    third.       The 
symbols  are,  therefore,  (a  :  a  :  cca),  OoO,   j  1 10}. 
figure  49,   each  face  is   a  rhombus  and,    hence, 
termed  the  rhombic  dodecahedron. 

The  crystallographic  axes  pass  through  the  tetrahedral  angles, 
the  trigonal  axes  join  opposite  trihedral  angles,  and  the  binary  axes 
the  centers  of  opposite  faces.  It  follows,  therefore,  that  the  faces 
are  parallel  to  the  secondary  planes  of  symmetry. 

3.  Hexahedron  or  Cube.     The  faces  of  this  form  cut  one  axis 
and    are  parallel  to    the  other  two.     This  is  expressed  by  (a  :  do  a 


Fig.  48. 


Fig.  49. 

In  the  ideal  form, 
the    form  is  often 


1)  The  heavy  lines  indicate  edges  through  which  principal  planes  of  symmetry  pass.  The  lighti 
full  lines  show  the  location  of  the  secondary  planes,  while  dashed  lines  indicate  the  absence  of  planes, 
3ee  page  26. 


20 


CUBIC   SYSTEM. 


Fig.  60. 


ooOoo,  jioo}.  Six  such  faces 
are  possible  and  when  the  development 
is  ideal,  figure  50,  each  is  a  square. 

The  crystallographic  axes  pass 
through  the  centers  of  the  faces.  The 
trigonal  axes  of  symmetry  join  opposite 
trihedral  angles,  while  the  binary  axes 
bisect  the  twelve  edges.  Compare  figures 
44,  45,  and  46. 


Fig.  51. 


Fig.  52. 


4.  Trigonal  trisoctahedron.1}     The  faces  of  this  form  cut  two 
crystallographic    axes    at    equal    distances,    the    third    at    a   greater 

distance  via.  The  coeffi- 
cient m  is  some  rational 
value  greater  than  one 
but  less  than  infinity.  The 
ratio  is  (a  :  a  :  via)  and 
it  requires  twenty -four 
such  faces  to  enclose 
space.  The  Naumann 
and  Miller  symbols  are 
mO,  \hhl\,  where  h  is 

greater  than  /.  Because  the  general  outline  of  this  form  is  similar 
to  that  of  the  octahedron,  each  face  of  which  in  the  ideal  forms  is 
replaced  by  three  equal  isosceles  triangles,  it  is  termed  the  tri- 
gonal trisoctahcdron,  figures  51,  20 1 22 1 5,  and  52,  3O\33i\. 

The  crystallographic  axes  join  opposite  octahedral  angles.  The 
trigonal  axes  pass  through  the  trihedral  angles  and  the  six  binary  axes 
bisect  the  twelve  long  edges. 

5.  Tetragonal  trisoctahedron.2)    This  form  consists  of  twenty- 
four  faces,  each  cutting  one  axis  at   a  unit's  distance  and  the  other 
two  at  greater  but  equal  distances  ma.     The  value  of  m  is,  as  above, 
m  >  i  <  oo.     The    symbols    are,    therefore,    (a  :  ma  :  ma),   mOm, 
\hll\,  h  >  I.       The    ideal     forms,    figures   53,    2O2J2ii(,    and   54, 
4O4|4iiJ,  bear   some    resemblance  to  the  octahedron,  each  face  of 
which  has  been  replaced  by  three  four-sided  faces,  trapeziums,    of 


!)    Also  known  as  the  trisoctahedron. 

2)    Also  termed  the  trapezohedron,  icositetrahedron,  andleucitohedron. 


HEXOCTAHEDRAL    CLASS. 


21 


equal  size.     The  form  is,   therefore,  termed  the  tetrag-onal  trisoc- 
tahedron.     The   six  tetrahedral  angles1)  a  indicate  the  position  of 
the       crystallographic 
axes.      The    trigonal 
axes  of  symmetry  join 
opposite     trihedral 
angles,   while  those    of 
binary   symmetry  con- 
nect   the    tetrahedral 
angles2)  b. 

6.     Tetrahexahe-  Fig.  53.  Fig.  54. 

dron.     In  this  form  the 

faces  cut  one  axis  at  a  unit's  distance,  the  second  at  the  distance 
ma,  where  m>\<  00,  and  extend  parallel  to  the  third  axis.  The 
symbols  are,  therefore, 
(a  :  ma  :  do  a),  oo  O  m, 
\hko\.  The  twenty- 
four  faces  in  the  ideal 
forms,  figures  55, 
ooO2[2ioS,  and  56, 
OoO4| 41  o|,  are  equal 
isosceles  triangles. 
Since  this  form  may  be 


Fig.  55. 


Fig.  56. 


considered  as  a  cube,  whose  faces  have  been  replaced  by  tetragonal 
pyramids,  it  is  often  called  the  pyramid  cube  or  tetrahexahedron. 
The  crystallographic  axes  are  located  by  the  six  tetrahedral 
angles.  The  axes  of  trigonal  symmetry  pass  through  opposite  hex- 
ahedral  angles,  while  the  binary  axes  bisect  the  long  edges. 

7.  Hexoctahedron. 
As  is  indicated  by  -the 
name,  this  form  is  bound- 
ed by  forty-eight  faces. 
Each  cuts  one  crystallo- 
graphic axis  at  a  unit's 
distance,  the  other  two  at 
greater  but  unequal  dis- 


Fig.  57. 


Fig.  58. 


1 )  With  four  equal  edges. 

2)  These  have  two  pairs  of  equal  edges. 


22 


CUBIC    SYSTEM. 


tances  na  and  ma,  respectively;  n  is  less  than  m,  the  value  of  m 
being,  as  heretofore,  w>i<oc.  Hence,  the  symbols  may  be 
written  (a:  na  :  ma\  mOn,  \hkl\.  Figures  57,  301)321  J,  and 
58,  5O|J53i  5,  show  ideal  forms,  the  faces  being  scalene  triangles 
of  the  same  size. 

The  crystallographic  axes  pass  through  the  octahedral  angles, 
while  the  hexahedral  angles  locate  those  of  trigonal  symmetry.  The 
binary  axes  pass  through  opposite  tetrahedral  angles. 

The  seven  forms  just  described  are  the  only  ones  possible  in  this 
class.  They  are  often  called  simple  forms. 

The  following  table  gives  a  summary  of  their  most  important 
features. 


FORMS 

SYMBOLS 

o* 
*£ 
11 

*£ 

8 

Number  of  Solid 
Angles 

Trihedral 

jt 

25 
tJ-a 

H 

i  1  Hexahe- 
dral 

I  1  .  |  Octahe- 
1  |  '  1  dral 

Weiss 

Naumann 

Miller 

Octahedron 

a  :  a  :  a 

0 

{ill} 

- 

6 

Dodecahedron 

a  :  a  :  Qoa 

acO 

\IIO\ 

12 

8 

6 

- 

Hexahedron 

a  :  oo  a  :  00  a 

oo  O  oc 

{IOO} 

6 

8 

— 

- 

- 

Trigonal 
trisoctahedron 

a  :  a  :  ma 

mO 

\hhl\ 

24 
24 

8 

— 

- 

6 

Tetragonal 
trisoctahedron 

a  :  ma  :  ma 

m  Om 

\hll\ 

8 

61>+12*> 

- 

- 

Tetrahexahedron 

a  :  ma  :  oc  a 

oo  Om 

\hko\ 

24 

- 

6 

8 

- 

Hexoctahedron 

a  :  na  :  ma 

mOn 

\hkl\ 

48 

- 

12 

8 

6 

From  this  tabulation  we  see  that  the  ratios  of  the  octahedron, 
dodecahedron,  and  hexahedron  contain  no  variables  and,  hence,  each 
is  represented  by  but  one  form.  These  are  often  called  singular  or 
fixed  forms.  The  other  ratios,  however,  contain  either  one  or  two 
variables  and,  therefore,  each  represents  a  series  of  forms.  Compare 
figures  51  to  58. 


!)    These  have  four  equal  edges. 

2)    Two  pairs  of  two  equal  edges  each. 


HEXOCTAHEDRAL     CLASS. 


23 


The   relationship   existing   between   the   simple   forms    is   well 
expressed  by  the  following  diagram: 


a  :  a  :  a 


a  :  a  :  oo  a 


a  :  na  :  ma 


a  :  ma  :  oo  a 


a  :  oca  :  ooa 


The  three  fixed  forms  are  placed  at  the  corners  of  the  triangle 
and,  as  is  obvious,  must  be  considered  as  the  limiting  forms  of  the 
others.  For  example,  the  value  of  m  in  the  trigonal  trisoctahedron 
(a  :  a  :  md)  varies  between  unity  and  infinity,  page  20.  Hence,  it 
follows  that  the  octahedron  and  dodecahedron  are  its  limiting  forms. 
The  tetragonal  trisoctahedron  (a  :  via  :  md)  similarly  passes  over 
into  the  octahedron  or  cube,  depending  upon  the  value  of  m.  The 
limiting  forms  are,  therefore,  in  every  case  readily  recognized. 
Those  forms,  which  are  on  the  sides  of  the  triangle1^  lie  in  the  same 
zone,  that  is,  their  intersection  lines  are  parallel. 

Combinations.  The  following  figures  illustrate  some  of  the 
combinations  (page  7)  of  the  simple  forms,  which  are  observed  most 
frequently. 


Fig.  59.  Fig.  60.  Fig.  61. 

Figures  59  and  60.     h  =  oo  O  oo,{ioo};  o=O,  {in}.     Com- 

1)    For  example,  the  octahedron,  trigonal- trisoctahedron,  and  dodecahedron. 


24 


CUBIC   SYSTEM. 


monly  observed  on  galena,  PbS.  In  figure  59  the  octahedron 
predominates,  whereas  in  figure  60  both  forms  are  equally 
developed. 

Figure  61.     h  =  00  O  oc,  {100} ;     d  =  ooO,  {no}.      Copper, 
Cu,  and  fluorite,  CaF2 ,  show  this  combination. 


Fig.  62. 


Fig.  63. 


Fig.  64. 


Figures  62  and  63.  h  =  ooO  00,{  100};  o  =  O,{iu};  and 
d  -=  ocO,  {no}.  Also  observed  on  galena,  PbS. 

Figure  64.  h  =  OoOoo,jioo|;  e  =  OoO2,  {210}.  Observed 
on  copper,  Cu;  fluorite,  CaF2 ;  and  halite,  NaCl. 


Fig.  65. 


Fig.  66. 


Fig.  67. 


Figure  65.     o  =  O,  { 1 1 1 } ;    e  =  0062,  J2io}.     Fluorite,  CaF2. 
Figure  66.     d=  ooO, {no};  e  =  0062,  J2io}.     Copper,  Cu. 
Figure  67.     h  =  oo  O  oo,{iooj ;   i  =  2O2,\2ii\.    Observed  on 
analcite,  NaAl(SiO3)2.H2O;  and  argentite,  Ag2S. 

Figure68.     o  =  O,  { 1 1 1 } ;  *' =  303, 13"}-     Spinel,  MgAl2O4. 


HEXOCTAHEDRAL   CLASS. 


Fig.  68. 


Fig.  69. 


Fig.  70. 


Figure69.     ,'=  2O2,{2ii};    O  =  O,{in}.     Argentite,  Ag2S. 
Figure  70.     o  =  0f  {in};     ^=ooO,  {nof;     f=2O2,{2ii 
Observed  on  spinel,  MgAl2O4,  and  magnetite,  Fe3O4. 


Fig.  71. 


Fig.  72. 


Fig.  73. 


Figure  71.     rf=ooO,{no};    f  =  2O2,{2ii{.      A  common  com- 
bination  observed  on  the  garnet,  R'^'^SijO  „. 

Figure  72.     ,'=202,{2ii{;    /  =  |O,{332}-     Garnet. 
Figure  73.     d=  ooO,{noJ;  5  =  3O|,J32i}.     Garnet. 


Figure  74.     rf= 
Garnet. 


Fig.  74. 
o};    /  =  2O2,{2ii(;    ^  = 


26 


CUBIC    SYSTEM. 


2.    HEXTETRAHEDRAL    CLASS.1) 

( Tetrahedral  Hemihedrism. ) 

Elements  of  Symmetry,  By  decreasing  the  elements  of  sym- 
metry of  the  hexoctahedral  class,  page  17,  the  other  classes  of  the 
cubic  system  may  be  deduced.  In  this  class  the  principal  planes  of 
symmetry  disappear,  which  necessitates  not  only  the  loss  of  the  six 

axes  of  binary  symmetry,  but  also  the  three  of 
tetragonal  symmetry  parallel  to  the  crystal- 
lographic  axes,  see  page  18.  The  center 
of  symmetry  also  disappears.  Hence,  the 
elements  remaining  are:  six  secondary  planes 
of  symmetry,  and  four  trigonal  axes,  which  are 
now  polar.  There  are  <also  three  binary  axes 
parallel  to  the  crystallographic  axes,  which 
in  the  hexoctahedral  class  are  of  tetragonal 
symmetry.  The  secondary  planes  of  sym- 
metry are  easily  located  since  they  pass  through  the  edges  of  the 
various  forms  of  this  class.  For  the  symmetry  relations  see  figure  75. 

Tetrahedral  Hemihedrism.  The  various  forms  of  this  class 
may  be  derived  from  the  holohedral  forms  by  allowing  all  faces  which 
occur  in  alternate  octants  to  be  suppressed  while  the  others  are 
extended.  This  is  illustrated  by  figure  75.  A  careful  study  of  this 
figure  shows  the  loss  and  retention  of  the  elements  of  symmetry  as 
outlined  in  the  preceding  paragraph. 

Tetrahedron.  By  allowing  alternate  faces  of  the  octahedron 
to  be  extended,  a  new  form  bounded  by  four  equilateral  triangles, 
intersecting  at  an  angle  of  78°  31'  44",  results.  This  form  is  the 

70* 


Fig.  75. 


Fig.  76.  Fig.  77. 

1)     The  tttrahedral group  of  Dana. 


Fig.  78. 


HEXTETRAHEDRAL    CLASS.  27 

tetrahedron.  Each  octahedron,  however,  yields  two  new  correlated 
forms  as  shown  by  figures  76,  77  and  78.  If  the  shaded  faces 
of  figure  77  are  suppressed,  we  obtain  the  tetrahedron,  whose 
position  is  indicated  by  figure  78.  However,  when  the  shaded 
faces  are  allowed  to  expand,  the  others  being  suppressed,  another 
tetrahedron,  figure  76,  similar  in  every  respect  to  the  first,  but  rotated 
through  an  angle  of  90°  is  obtained.  These  forms  are,  hence, 
congruent  and  differentiated  by  the  plus  and  minus  signs,  figure 
78,  plus  or  positive;  figure  76,  minus  or  negative.  The  symbols 
of  the  tetrahedron  are  written: 

a:  a:  a         O  A.  O 

4_ — ;  -f  —  , KJIII;  };  and  —  —  K{III[. 

The  crystallographic  axes  pass  through  the  centers  of  the  edges. 
The  axes  of  binary  symmetry  are  parallel  to  them;  those  of  trigonal 
symmetry  pass  from  the  trihedral  angles  to  the  centers  of  opposite 
faces. 

Tetragonal  tristetrahedron.  From  the  trigonal  trisoctahedron, 
as  shown  by  figures  79,  80,  and  81,  are  obtained  two  congruent 
forms  bounded  by  twelve  faces,  which  in  the  ideal  development  are 


Fig.  79.  Fig.  80.  Fig.  81. 

similar  trapeziums.     These    forms  possess  a  tetrahedral    habit  and 
are  called    tetragonal    tristetrahedrons,   sometimes   also   deltoid 
dodecahedrons  or  simply  deltoids. 
The  symbols  are: 

a:  a:  ma            mO       ,        7>   f                             mO 
-f-   — ;     -f ,  K\hhl\  (figure  80;  ,    *\hhl\ 

(figure  79). 


1)  The  letfer  K  (KA.O/OS,  inclined)  is  placed  before  the  Miller  indices  of  forms  of  this  class 
because  their  faces  are  inclined.  This  type  of  hemihedrism  is  often  known  as  the  inclined-face 
hemihedrism. 


28  CUBIC   SYSTEM. 

The  crystallographic  axes  pass  through  opposite  tetrahedral 
angles,  while  the  axes  of  trigonal  symmetry  join  opposite  trihedral 
angles,  one  of  which  is  acute,  the  other  obtuse. 

Trigonal  tristetrahedron.  The  application  of  the  tetrahedral 
hemihedrism  to  the  tetragonal  trisoctahedron  produces  two  congruent 
half-forms,  bounded  by  twelve  similar  isosceles  triangles,  figures* 
82,  83,  and  84.  Since  these  new  forms  may  be  considered  as 
tetrahedrons  whose  faces  have  been  replaced  by  trigonal 
pyramids,  they  are  termed  trigonal  tristetrahedrons^  or  pyramid 
tetrahedrons.  Sometimes,  moreover,  the  term  trigonal 
dodecahedron  is  also  used. 


Fig.  82.  Fig.  83.  Fig.  84. 

The  symbols  are  written: 

a \ma\ma      ,  mQm  mOm          -. 

+/      -T-T-5    +  -T~>   «{A//}  (figure  84);-      -j-,  «{*//} 

(figure  82). 

The  crystallographic  axes  bisect  the  long  edges.  The  trigonal 
axes  pass  from  the  trihedral  angles  to  the  opposite  hexahedral. 

Hextetrahedron.  In  precisely  the  same  manner  the  hexoctahe- 
dron,  figure  86,  yields  two  congruent  forms  of  tetrahedral  habit, 
bounded  by  twenty-four  similar  scalene  triangles.  These  new  forms 
are  called  hextetrahedrons. 


*)  The  following  relationship  between  this  and  the  preceding  form  and  their  respective 
holohedral  forms  should  be  carefully  noted,  viz:  The  trigonal  trisoct?hedron  (each  face  is  triangular) 
yields  the  tetragonal  tristetrahedron  (faces  of  tetragonal  outline),  wh'le,  vice  versa,  the  tetragonal 
trisoctahedron  furnishes  the  trigonal  tristetrahedron.  Compare  figures  79  to  84. 


HEXTETRAHEDRAL    CLASS. 


29 


Fig.  85. 
The  symbols  are: 


a  \na\ma        mQn 


\hkl\   (figure  87);  - 


hkl\ 


(figure  85.) 


The  crystallographic  axes  connect  opposite  tetrahedral  angles. 
The  trigonal  axes  of  symmetry  pass  through  opposite  hexahedral 
angles,  one  of  which  is  more  obtuse  than  the  other. 

Hexahedron,    dodecahedron,    and    tetrahexahedron.      As   is 

evident  from  figure  7 5,  no  forms,  geometrically  new,  can  result  from 
the  application  of  this  type  of  hemihedrism  on  the  cube,  dodecahedron, 
and  tetrahexahedron.  The  faces  of  these  forms  belong  simultane- 
ously to  different  octants  and,  hence,  the  suppression  of  a  portion 
of  a  face  in  one  octant  is  counterbalanced  by  a  corresponding 
expansion  in  another.  Therefore,  no  change  in  these  forms  can 
result.  They  are  from  a  geometrical  standpoint  exactly  similar  to 
those  of  the  hexoctahedral  class.  Their  symmetry  is,  however,  of  a 
lower  grade.  This  is  not  to  be  recognized  on  models.  On  crystals, 
however,  the  form  and  position  of  the  so-called  etch  figures,  as  also 
the  different  physical  characteristics  of  the  faces,  reveal  the  lower 
grade  of  symmetry1).  These  forms  are,  therefore,  only  apparently 
holohedral. 


!)    When   crystals   are    subjected   to    the   action  of   some  solvent    for  a  short   time,  small 
depressions   or    elevations,  the  so-called  etch  figures,    appear.     Being    dependent  upon  the  internal 
molecular  structure,  their  form  and  position  indicate  the  symmetry  of  the  crystal.     For  instance,  figure 
88  shows  the  etch  figures  on  a  crystal  (cube)  of  halite, 
NaCl.     Here,  it  is  evident,  that  the  symmetry  of  the 
figures    in  respect  to   that  of  the  cube  is  such  as  to 
place  the  crystal  in  the  hexoctahedral  class.     Figure 
81)  represents  a  cube  of  sylvite,  KC1,  which  geometri- 
cally   does  not  differ  from   the  crystal    of  halite.    A 
lower  grade  of   symmetry  is,  however,   revealed  by 
the  position  of  the  etch  figures.     This  crystal  belongs 
to  the  pentagonal  icositetrahedral  class,   page  36,   for 
no  planes  of  symmetry  can  be  passed  through  these  Fig  gg^ 


30 


CUBIC   SYSTEM. 


The  following  table  shows  the  important  features  of  the  forms 
of  this  class  of  symmetry. 


FORMS 

SYMBOLS 

Number  of 
Equal  Faces 

Number  of  Angles 

Trihedral 

Tetrahedral 

|  |  Hexahedral 

Weiss 

Naumann 

Miller 

Tetrahedron 

a:  a:  a 

\ 

+  2 

0^ 

I           2 

K\III\ 

K\HI\ 

K 

J 

4 



±-         2 

Tetragonal 
tristetrahedron 

a:  a:  ma 

mO 

*\hhl\ 
*\hhl\ 

1 

[l2 

4+4 

6 



,             2 

mO 

2 

I            2 

Trigonal 
tristetrahedron 

a  :  ma  :  ma 

\ 

(     mO?n 

i 

K\kll\ 
K\hJl\ 

\12 

4 

— 

4 

H          2 

mOm 

2 

(          2 

Hextetrahedron 

a:na:  ma 

f  ,    mOn 

K\hkl\ 
K\k~kl\ 

1 
\24 

\ 

— 

6 

4 

+ 
4 

1 

2 

mOn 

2 

2 

Dodecahedron 

a:  a:  oca 

ocO 

K{l  10} 

Apparently 
holohedral 

J 

Tetrahexahe- 
dron 

a  :  ma  ;  oc  a 

ocO;;/ 

*{hko} 

Hexahedron 

a:  oca:  cca 

ocO  oc 

K{IOO} 

Fig.  90. 


Fig.  91. 


figures,  which  at  the  same  time  are  planes  of 
symmetry  of  the  cube,  as  is  the  case  with  the 
crystal  of  halite,  figure  88.  Fora  fuller  discussion 
of  these  interesting  figures  consult  Groth,  Physi- 
kalische  Krystallographie,  4te  Auflage,  1905,  251; 
also  Dana,  Text-book  of  Mineralogy,  1898,  149. 

Figures  90  and  91  show  two  cubes  repre- 
senting crystals  of  sphalerite,  ZnS  (figure  90),  and 
pyrite,  FeS2  (figure  91).  Although  these  form? 
do  not  differ  from  the  holohedral,  it  is  at  once  seen 


HEXTETRAHEDRAL    CLASS. 


31 


Combinations.     Some  of  the  more  common   combinations  are 
illustrated  by  the  following  figures: 


Fig.  92. 


Fig.  93. 


Fig.  94. 


O 


Figure  92.     o  =  -\ ,    KJIIIJ;     h  =  OoO  00,  K|  iooj.         Ob- 
served on  sphalerite,  ZnS. 

Figure  93.     o  =  +  — ,  *{  1 11 } ;     o'  =  -      —  ,    K{III(. 


2 

O 


Sphalerite. 

Figure  94.     0=+—,    "Jin!;    h  =  ooO  oo,  *{  ioo[  ;    d 
00  O,  K{IIO{.     Tetrahedrite  (Cu2,  Fe,  Zn)4  (As,  Sb)2  S7. 


O 

Figure  95.       o  ~-     -h  —  , 


2O2 
=   -     -  ie2ii-. 


Tetrahedrite. 


Fig.  95. 

2O2 

Figure  96.     n  =  +       — , 


Fig.  96. 


Fig.  97. 


r  = 


ooO, 


Tetrahedrite. 


that  adjoining  faces  possess  striations  extending  in  different  directions.  Figure  90  possesses  six 
secondary  planes  of  symmetry,  the  principal  are  wanting,  and,  hence  this  cube  is  only  apparently 
holohedral.  It  belongs  to  hextetrahedral  class.  Figure  91  possesses  a  still  lower  grade  of  sym- 
metry for  here  only  three  principal  planes  are  present,  and  it  must,  therefore,  be  referred  to  the 
dyakisdodecahedral  class,  page  32, 


32 


CUBIC    SYSTEM. 


Figure  97.     d—   ooO,   K{IIO|;    m- 
Sphalerite. 

Figure  98.     /i=ooOoc,   K{IOO|;    o 


O 


,KJ3iiJ. 


+  — ,  K\\\\   o 


Fig.  98. 


— 


;    d  = 


z;  = 


c  O 


"^SSS1!*     This  combination  occurs  on  boracite,  Mg7Cl2B16O30. 

3.    DYAKISDODECAHEDRAL   CLASSY 

(^Pyritohedral  Hemihedrism.} 

Elements  of  Symmetry.  Here  the  six  secondary  planes  and 
the  six  axes  of  binary  symmetry  are  lost.  The 
elements  of  symmetry  which  remain  are,  there- 
fore, three  principal  planes  parallel  to  the 
planes  of  the  crystallographic  axes,  four 
trigonal  axes,  and  also  the  center  of  symmetry. 
The  three  axes  parallel  to  the  crystallographic 
axes  now  possess  binary  symmetry.  Figure  99 
shows  these  elements  as  well  as  the  application 
of  the  hemihedrism,  which  will  be  discussed 
in  the  next  paragraph. 


Fig.  99. 


Pyritohedral  hemihedrism  2.  In  this  class  the  forms  may  be 
assumed  as  derived  from  the  holohedral  types  by  the  expansion  of 
the  faces  lying  wholly  within  alternate  spaces  formed  by  the 
intersection  of  the  secondary  planes  of  symmetry,  page  18.  Such 

1)  Called  by  Dana  the  pyritohedral group. 

2)  Also  termed  pentagonal  or  parallel-face  hemihedrism,  having  reference  to   the  pentagonal 
outline  and  parallel  arrangement  of  the  faces,  respectively. 


DYAKISDODECAHEDRAL    CLASS. 


33 


faces  or  pairs  of  faces  are  then  symmetrical  to  the  three  principal 
planes  of  symmetry,  figure  99. 

Pyritohedron.  The  application  of  this  type  of  hemihedrism  to 
the  tetrahexahedron,  figure  101,  produces  two  correlated,  congruent, 
hemihedral  forms,  figures  100  and  102,  bounded  by  twelve  similar 
faces.  Each  face  is  an  unequilateral  pentagon,  four  sides  of  which 
are  equal.  Since  these  forms  are  congruent,  a  rotation  through  90° 
is  all  that  is  necessary  to  bring  them  into  precisely  the  same  position 
in  space.  They  are,  hence,  designated  as  plus  and  minus  forms. 1} 


Fig.  100. 

The  symbols  are: 
a:  ma\  do  a 


Fig.  101. 


Fig.  102. 


fa:ma:aca'}          f  ocOral 
±|        -J-       J;    +|  -J-   J,    TJA£0}     (figure 


102); 


(figure  100). 


Because  this  form  occurs  frequently  on  pyrite  it  is  termed  the 
pyritohedron,  although  pentagonal  dodecahedron^  is  quite  often 
used. 

The  axes  of  binary  symmetry,  hence,  also  the  crystallographic 
axes  bisect  the  six  long  edges.  The  trigonal  axes  pass  through  the 
trihedral  angles,  the  edges  of  which  are  of  equal  lengths. 


1)  Also  designated  as  right  and  left.    It  is  preferable,  however,  to  use  those  terms  in  reference 
to  enantiomorphous  forms  only,  page  16. 

2)  In  this  class,  TT  (TrapoAAryAos,  parallel  faced)  is  placed  before  the  Miller  indices,  see  page 
27.    To  distinguish  the  other  symbols  from  those  of  the  preceding  class,  they  are  enclosed  in  brackets. 

3)  The  regular  pentagonal   dodecahedron  of   geometry,    bounded  by  equilateral    pentagons 
intersecting  in  equal  edges  and  angles  is  crystallographically  an  impossible  form,  the  value  of  m  being 


which  is  irrational,  compare  page  10. 


^4  CUBIC   SYSTEM. 

Dyakisdodecahedron.  By  the  expansion  and  suppression  of  alter- 
nate pairs  of  faces  crossed  by  the  principal  planes  of  symmetry,  the 
hexoctahedron,  figure  104,  yields  two  correlated,  congruent  forms, 
each  bounded  by  twenty-four  similar  trapeziums,  figures  103  and  105. 
These  forms  are  the  dyakisdodecahedrons%  also  termed  didodecahe- 
drons,  or  diploids. 


Fig.  103. 


Fig.  104. 


Fig.  105. 


The  symbols  are 

\a:na:ma 
±|_         2  -- 

ir\hlk\  (figure  103). 


'   -«\hkl\  (figure  105):— 


mOn 


The  crystallographic  axes,  also  those  of  binary  symmetry,  pass 
through  the  six  tetrahedral  angles  possessing  two  pairs  of  equal  edges. 
The  trigonal  axes  join  opposite  trihedral  angles. 

Other  Forms.  The  hexahedron,  octahedron,  dodecahedron, 
trigonal  trisoctahedron,  and  tetragonal  trisoctahedron,  the  other 
forms  of  the  holohedral  type,  do  not  yield  new  forms  by  the  applica- 
tion of  this  hemihedrism.  This  is  clearly  shown  by  figure  99,  illus- 
trating the  elements  of  symmetry  of  this  class.  Being  apparently 
holohedral,  these  forms  may  occur  independently  or  in  combination, 
but  always  with  the  full  number  of  faces.  Their  symmetry  is,  how- 
ever, less  than  that  of  the  holohedrons.  See  page  29. 


In  the  following  table -the  important  features  of  the  forms  of 
this  class  are  indicated. 


DYAKISDODECAHEDRAL    CLASS. 


35 


FORMS 

SYMBOLS 

Number  of  Faces 

Number  of 
Angles 

Trihe- 
dral 

Tetra- 
hedral 

"«  en 

3   V 
0"&C 

'^ 

rnS 

+£ 

<Mhti 

rer 
3he 

°*E 

NW 

+  s 

I-H  bi 

+s 

N  

ll 

Weiss 

Naumann 

Miller 

Octahedron 

a  :  a  :  a 

o 

TTJ/J/j 

AI 

j 

ppa 
hoi 

itly 
dra 

Dodecahedron 

a  :  a  :  oca 

ooO 

TT\IIO\ 

Hexahedron 

a  :  oca  :  oca 

oo  O  oo 

7T  j/OOJ 

Trigonal  trisoc- 
tahedron 

a  :  a  :  ma 

mO 

*\hhl\ 

Tetragonal  tris- 
octahedron 

a  :  ma  :  ma 

mOm 

*\hll\ 

Pyritohedron 

&Sfrs 

[a'-ma'-aoc  1 

\  \  r°°0wi 

*\hko\ 
*\kho\ 

\ 

8 

12 

— 

+  L       2       J 

rooOwzi 

2            \ 

,       L        2       1 

Dyakisdodeca- 
hedron 

ra:  na  :ma~\ 

TmOn-t 

•*\hkl\ 
*\hlk\ 

24 

8 

- 

6 

12 

j  '  L  ^  J 

[wOw"| 

2     J 

,    -  J 

Combinations.     The  accompanying  figures  show  some  combina- 
tions of  the  forms  of  this  class. 


Fig.  106. 

Figure  1 06.       h  =  ocO  oo,  v\  100} ; 
Commonly  observed  on  pyrite,  FeS2. 
Figures  107  and  108.    o  =  O,  ir\  i 1 1  \ ; 


+ 


1        g     J, 


7r2iOi 


36 


CUBIC   SYSTEM. 


In  figure  107  the  octahedron  predominates,  whereas  in  figure  108 
both  forms  are  about  equally  developed.     Pyrite. 


Fig.  109.  Fig.  110.  Fig.  111. 

Figure    109.   h  =ooOoo,  TrjiooJ;    5  =  +  I  ^—M,  w|32i|. 
Pyrite. 

Figure  no.     e=  -f  I,   7r|2ioS;  o  =  O,  TT{III\. 

Between  o  and  e  but  in  the  same  zone,    -M  I  ^SS21 1-      Pyrite. 

Figure   in.      ^  =   ooOoo,     TT\\QQ\\     o  =    O,  TT|  1 1 1 1 ;    e  = 

[ooO2l 
I  7r|2io|.     Cobaltite,  CoAsS. 


4.    PENTAGONAL  ICOSITETRAHBDRAL  CLASS.1) 

( Plagihedral  or  Gyroidal  Hemihedrism. ) 

Elements  of  Symmetry.     All  planes,  as  well  as  the  center  of 
symmetry,  are  lost.     There  remain,  however,  all  of  the  thirteen  axes, 

namely,  three  tetragonal,  four  trigonal,  and 
six  binary  axes.  Figure  112  shows  these 
elements  and  also  the  method  of  the  applica- 
tion of  the  plagihedral  hemihedrism. 

Plagihedral  Hemihedrism.     Forms    of 
this  class  may    be  assumed  as   derived  from 
the  holohedrons  by   the    alternate  expansion 
and  suppression  of  faces  lying  wholly  within 
each  of  the  forty-eight  sections,  resulting  from 
,    the  intersection    of  the  nine  planes  of  sym- 
metry, page  1 8.      Compare  figure  112. 


Fig.  112. 


Termed  by  Dana  the  plagihedral  group. 


PENTAGONAL    ICOSITETRAHEDRAL    CLASS. 


37 


Pentagonal  Icositetrahedron.  By  the  extension  and  suppres- 
sion, respectively,  of  alternate  faces  of  the  hexoctahedron,  figure  114, 
two  new  correlated  forms,  bounded  by  twenty-four  equal  unsymmet- 
rical  pentagons  result.  *  These  forms  are  termed  pentagonal  icositet- 
rahedrons  or  gyroids.  They  are  not  congruent,  for  they  cannot  be 
brought  into  the  same  position  by  rotation  and,  hence,  are  not 
superimposable.  Therefore,  they  are  enantiomorphous,  page  16, 
and  distinguished  as  right,  figure  115,  and  left,  figure  113,  accord- 
ing to  which  of  the  top  faces  (right  or  left)  in  the  front  upper  octant 
has  been  extended. 


Fig.  113. 


The  symbols  are: 


Fig.  114. 


Fig.  115. 


a:  na  :  ma      mOn 
r.l-  -\r  --  * 


mOn 


The  crystallographic  axes,  also  those  of  tetragonal  symmetry, 
join  opposite  tetrahedral  angles.  The  trigonal  axes  pass  through 
those  trihedral  angles  which  possess  equal  edges,  while  the  six  binary 
axes  bisect  the  twelve  edges  intermediate  between  those  forming  the 
tetrahedral  angles. 

Other  forms.  ^Jhe  six  other  holohedrons  are  not  from  a  geo- 
metrical standpoint  effected  by  this  hemihedrism.  A  study  of  figure 
112  shows  that  no  new  forms  can  result,  for  on  these  holohedrons 
none  of  the  faces  lie  wholly  within  one  of  the  forty-  eight  sections 
referred  to  on  page  36.  The  octahedron,  dodecahedron,  hexahedron, 
the  trigonal  and  tetragonal  trisoctahedrons,  and  the  tetrahexahedron 
are  only  apparently  holohedral. 

i)  In  this  class,  y  (yvpos,  bent)  is  placed  before  the  Miller  indices.  Compare  footnotes  on 
pages  27  and  33. 


38 


CUBIC    SYSTEM. 


The  principal  features  of  the  forms  of  this  class  may  be  tabulated 
as  follows: 


FORMS 

SYMBOLS 

Number  of  Faces 

Number  of  Angles 

Trihedral 

Tetrahedral 

I! 

WS 

« 

"i  « 

ii 

D  W 
eo 

Weiss 

Naumann 

Miller 

Octahedron 

a  :  a  \a 

0 

y\in\ 

1 

App 

he 

aren 
)lohe 

tly 
^dral 

Dodecahedron 

a  :  a  :  oca 

oc<9 

y\fio\ 

Hexahedron 

a  :  oc  :  oca 

oo  O  oo 

y\ioo\ 

Trigonal 
trisoctahedron 

a  :  a  :  ma 

mO 

y\hhl\ 

Tetragonal 
trisoctahedron 

a  :  ma  :  ma 

m  Om 

y\hll\ 

Tetrahexahe- 
dron 

a  :  ma  :  oca 

oc  Om 

y\hko\ 

Pentagonal 
Icositetra- 
hedron 

a\ita\  ma 

7ii  On 
mOn 

y\khl\ 
y\hkl\ 

24 

8 

24 

6 

r,l         2 

2 

Combinations.  Figure  116.  2=202, 
y\2ii\mt.s  =/ — ~>y{%75l-  Sal  Ammoniac, 
NH4C1. 

5.     TBTRAHEDRAL  PENTAGONAL  DODECAHE- 
DRAL   CLASSY 

(  Tetartohedrism. ) 

Fig.  116.  Elements    of    Symmetry.        The     nine 

planes,  the  three  axes  of  tetragonal  and  the  six 

of  binary  symmetry,   as  also  the  center,  are  lost.     The  elements  of 
this  class  are,  hence,  four  polar  axes  of  trigonal  and  three  of  binary 


1)     The  tetartohedral group  of  Dana. 


TETRAHEDRAL  PENTAGONAL  DODECAHEDRAL  CLASS. 


39 


symmetry,  the    latter    being  parallel    to    the 
crystallographic  axes.     See  figure  117. 

Tetartohedrism.  The  simple  tetartohe- 
dral  forms  may  be  conceived  as  derived  from 
the  hemihedrons  of  any  one  of  the  three 
hemihedral  classes — tetrahedral,  pyritohedral, 
or  plagihedral  —  by  the  subsequent  applica- 
tion of  one  of  the  other  two  methods.  Th^ 
result  is  in  each  case  the  same.  Compare 
figures  118,  119,  120,  122,  and  125. 


Fig.  117. 


Fig.  121. 


Fig.  122. 


Fig.  123. 


Fig.  124. 


Fig.  126. 


Fig.  125. 

Tetrahedral    pentagonal  dodecahedron.     From  the  hexoctahe- 
dron  by  the  successive  application  of  two  types  of  hemihedrism  four 


40  CUBIC    SYSTEM. 

new  forms  of  tetrahedral  habit,  bounded  by  twelve  unsymmetrical 
pentagons,  result.  The  derivation  of  these  forms  is  easily  to  be  seen 
from  figures  121-126,  also  figure  37,  page  15.  In  figure  122, 
the  form  derived  by  the  expansion  of  the  unshaded  faces  is  known  as 
the  left  positive  tetrahedral  pentagonal  dodecahedron,  whereas 
the  one  derived  from  the  shaded  faces  alternating  with  these  —  i.  e., 
in  the  same  octant  —  is  the  rig-Jit  positive  form.  From  figure  125 
the  left  negative  and  rig-Jit  negative  forms  are  likewise  possible. 

7    a\  na\  ma 

The  symbols  according  to   Weiss   are    -r     r,  -      I . 

4 
According  to  Naumann  and  Miller  they  may  be  written: 

1.  -\-  r  -   — -,  Kir\khl\*\  Positive  right,  figure  123. 

til  O  n 

2.  — r  -    — ,  KTr\hkl\,  Negative  rig  Jit,  figure  126. 

4 

3.  -f-  /-     — ,  KTr\hklt\,  Positive  left,  figure  121. 

4 

4.  -/-     — ,  KTr\khl\,  Negative  left,  figure  124. 

Forms  i  and  2,  also  3  and  4,  are  among  themselves  congruent, 
whereas  the  pairs  I  and  3,  and  2  and  4,  are  enantiomorphous. 

The  crystallographic  axes  bisect  the  six  equal  edges,  while  the 
four  trigonal  axes  join  opposite  trihedral  angles  of  which  one  is  more 
obtuse  than  the  other.  These  trihedral  angles  possess  equal  edges. 

Other  forms.  If  tetartohedrism  be  applied  to  the  other  holohe- 
drons,  it  follows  that  four  of  them  will  yield  forms,  as  follows: 

Octahedron  —  two  tetrahedrons,  figures  76  and  78,  page  26. 

Trigonal  trisoctahedron  —  two  tetragonal  tristetrahedrons,  figures 
79  and  81,  page  27. 

Tetragonal  trisoctahedron  —  two  trigonal  tristetrahedrons,  figures 
82  and  84,  page  28. 

Tetrahexahedron  —  two    pyritohedroris,    figures    100   and    102, 

page  33- 

The  first  three  forms  do  not  differ  geometrically  from  those 
obtained  by  the  tetrahedral  hemihedrism,  while  the  fourth  is  the  same 
as  that  derived  by  the  pyritohedral  hemihedrism.  They  are,  hence, 

i)  Two  letters,  each  representing  one  of  the  types  of  hemihedrism  applied,  are  used  with  the 
Miller  indices  in  this  class. 


TETRAHEDRAL  PENTAGONAL  DODECAHEDRAL  CLASS. 


41 


only  apparently  hemihedral  forms.     The  cube  and  dodecahedron  are 
unchanged  geometrically  and,  therefore,  apparently  holohedral. 

The  following  table  shows  the  principal  features  of  the  forms  of 
this  class: 


FORMS 

SYMBOLS 

Number  of 
Faces 

Number  of 
Angles 

\Veiss 

Naumann 

Miller 

Trihedral 

Equal 
Edges 

Unea4ual 
Edges 

Dodecahedron 

a  :  a  :  oo  :t 

oc  O 

KTT\IIO\ 

"j 

i  Apparently 
holohedral 

Hexahedron 

a\  oca:  GO  .7 

oo  O  oo 

KTT\IOO\ 

Tetrahedrons 

a:  a:  a 

I           0 

l  ^ 

0 

2 

KTT\III\ 
KTT\III\ 

1 

Ap 
t( 
h 

j 

parent 
^trahec 
emihec 

iy 

Iral 
Irons 

i-     2 

Tetragonal 
tristetrahe- 
drons 

a:a:  ma 

i 

^mO 

KK\hhl\ 

**\hhl\ 

2 

mO 

2 

2 

Trigonal 
tristetrahe- 
drons 

a  :  ma  :  ma 

m  Om 

««\hll\ 

KTT\hn\ 

1 

2 

m  Om 

t«           2 

2 

Pyritohedrons 

[a  :  ma  :  oo  a  ~| 

'  ,focOm^ 

KTT\/lkO\ 
KTT\khO\ 

1 
Apparently 
}-    pyritohedral 
hemihedrons 

L   -  J 

roc<9w~i 
i  L  *  J 

2             J 

Tetrahedral 
pentagonal 
dodecahe- 
drons 

a:na:  ma 

mOn 

4 

mOn 
—  r  — 

4 
.    7mOn 

™\khl\ 

KTT\hkl\ 
KTT\hkl\ 

M\khl\ 

1 

12 

4  +  4J) 

12 

4 
±r,±J 

\  I 
4 

mOn 

L           4 

1 )     One  set  is  more  obtuse  than  the  other. 


42 


CUBIC    SYSTEM. 


Combinations.     In  this  class  it  is  evident  that  the  apparently 
holohedral  and  hemihedral  forms  of  the  various  classes,  as  described 

on  pages  40  and  41,  may 
occur  together  on  the  same 
crystal.  For  example, 
figures  127  and  128  show 
crystals  of  sodium  chlorate, 
NaClO3,  where  h  =  00  O  oo, 


10 


Fig.  127. 


Fig.  128. 


=    OCO,   KIT 

O 

K7T  |  I  I  I 


(figure  127)  =      -  [  ^-2],  „{  I20}  .p  (figure  I2g)=  +  F  GC021 

\  "  ^* 


Fig.  129. 


Fig.  130. 


Fig.  131. 


R  rtsrn  N  AI3°'  and   ISI  illustrate  crystals   of  barium  nitrate, 

Ba(N03)2,  with  the  following  forms:     h  =  --   ooO  oo,   KTTJ  ioo; ;    o  = 

--°.    wir7Tl.^_  T^02 

2 


X= 


I  «{«!{;#=  -^-|,    WjI20|; 


T-'l 


• 


HEXAGONAL  SYSTEM. 


-c 

Fig.  132. 


Crystallographic  Axes.  This  system  includes  all  forms  which 
can  be  referred  to  four  axes,  three  of  which  are  equal  and  lie  in  a 

horizontal  plane,  and  intersect  each  other 
at  an  angle  of  60°.  These  are  termed  the 
secondary  or  lateral  axes,  being  designated 
by  the  letter  a.  These  axes  are  inter- 
changeable. The  fourth,  a  principal  axis, 
is  perpendicular  to  the  plane  of  the  second- 
ary axes  and  is  termed  the  c  axis.  It  may 
be  longer  or  shorter  than  the  secondary 
axes.  The  three  equal  axes,  which  bisect 
the  angles  between  the  secondary  axes,  are 
the  intermediate  axes.  These  may  be 
designated  by  b.  Figure  132  shows  an  axial 
cross  of  this  system. 

In  reading  crystals  of  the  hexagonal  system,  it  is  customary  to 
hold  the  c  axis  vertical,  letting  one  of  the  secondary  or  a  axes  extend 
from  right  to  left.  The  extremities  of  the  secondary  axes  are 
alternately  characterized  as  plus  and  minus,  see  figure  132.  In 
referring  a  .form  to  the  crystallographic  axes,  it  is  common  practice 
to  consider  them  in  the  following  order:  #t  first,  then  a2,  thirdly  «3, 
and  lastly  the  c  axis.  The  symbols  always  refer  to  them  in  this 
order.  It  is  also  to  be  noted  that  in  following  this  order,  one  of  the 
lateral  axis  will  always  be  preceded  by  a  minus  sign. 

*  Since  the  lengths  of  the  a  and  c  axes  differ,  it  is  necessary  to 
assume  for  each  substance  crystallizing  in  this  system  a  fundamental 
form,  whose  intercepts  are  taken  as  representing  the  unit  lengths  of 
the  secondary  and  principal  axes,  respectively.  The  ratio,  which 
exists  between  the  lengths  of  these  axes  is  called  the  axial  ratio  and 
is  always  an  irrational  value,  the  a  axis  being  assumed  as  unity, 
page  ;•  '  [43] 


44 


HEXAGONAL    SYSTEM. 


Classes  of  Symmetry.  The  hexagonal  system  includes  a  larger 
number  of  classes  of  symmetry  than  any  other  system,  namely, 
twelve.  The  order  in  which  they  will  be  discussed  is  as  follows: 


i. 


tt 
t  * 
t* 


2. 

3- 
4- 
5- 
6. 

7- 
8. 

9- 

10. 

ii. 


(  Holohedrism}. 
f  Holohedrism  and\ 
\hemimorphism.      j 


( Hemihedrism). 

(Hemihedrism  and\ 
\hemimorphism.      J 

( Tetartohedrism.) 


12.     Trigonal  pyramidal  class 


Dihexagonal  bipyramidal  class 
Dihexagonal  pyramidal  class 

Ditrigonal  bipyramidal  class 
Ditrigonal  scalenohedral  class 
Hexagonal  bipyramidal  class 
Hexagonal  trapezohedral  class 
Ditrigonal  pyramidal  class 
Hexagonal  pyramidal  class 
Trigonal  bipyramidal  class 
Trigonal  trapezohedral  class 
Trigonal  rhombohedral  class 

Teta  rtohedrism 
and  hemimor- 

phism 

( Ogdohedrism). 
Those  classes  marked  with  an  *  are  the  most  important,  for 
nearly  all  of  the  crystals  of  this  system  belong  to  some  one  of  them. 
No  representatives  have  as  yet  been  observed  for  the  classes  marked 
by  t.  Those  marked  %  are  often  grouped  together  and  form  the 
trigonal  system. 

/.     DIHEXAQONAL  BIPYRAMIDAL   CLASS.U 

(Holohedrism.} 

Symmetry,  This  class  possesses  the  highest  grade  of  symmetry 
of  any  in  the  hexagonal  system. 

a)  Planes.  In  all  there  are  seven  planes  of  symmetry.  One  of 
these,  the  -principal  plane,  is  parallel  to  the  plane  of  the  secondary 
axes,  hence,  horizontal.  The  other  planes  are  divided  into  two 
series  of  three  each,  which  are  termed  the  secondary  and  inter- 
mediate, respectively.  Each  of  the  secondary  planes  includes  the  c 
or  principal  axis  and  one  of  the  secondary,  a,  axes.  These  planes 
are,  therefore,  vertical  and  perpendicular  to  the  principal  plane. 


The  normal  group  of  Dana. 


DIHEXAGONAL    BIPYKAMIDAL   CLASS. 


45 


Fig.  133. 


They  intersect  at  angles  of  60°.  The  inter- 
mediate planes  are  also  vertical  and  perpen- 
dicular to  the  principal  plane^,  for  they 
bisect  the  angles  between  the  secondary 
planes  and,  hence,  each  includes  the.c  and 
one  of  the  intermediate  axes. 

The  secondary  and  principal  planes 
divide  space  into  twelve  equal  parts,  called 
dodecants;  the  seven  planes,  however,  into 
twenty-four  parts,  figure  133. 

These  planes  are  often  designated  as 
follows: 

i  Principal  -f-  3  Secondary  -f  3  Intermediate  =  7  Plants. 

b)  Axes.     Parallel    to    the    vertical    or   c  axis    is  an    axis    of 
hexagonal  symmetry,  while  the  axes  parallel  to  the  secondary  and 
intermediate  axes  possess  binary  symmetry.     These  axes  are  often 
indicated,  thus, 

i*+3«-h3«  =  7  axes. 

c)  Center.      This  element  of  symmetry  is  also  present,  requiring 
every    face    to    have    a    parallel    counter-face.       Figure    134,    the 
projection    of    the  most  complicated  form 

upon  a  plane  perpendicular  to  the  vertical 
axis,  shows  the  elements  of  symmetry  of 
this  class. 


i.  Hexagonal  bipyramid  of  the  first 
order.  From  figure  132,  it  is  obvious  that 
any  plane  which  cuts  any  two  adjacent 
secondary  axes  at  the  unit  distance  from 
the  center  must  extend  parallel  to  the 
third.  If  such  a  plane  be  assumed  to  cut 
the  c  axis  at  its  unit  length  from  the  center,  the  parametral  ratio 
would  then  be 


Fig  134 


According  to  the  above  elements  of  symmetry,  twelve  planes 
possessing  this  ratio  are  possible.  They  enclose  space  and  give  rise 
to  the  form  termed  the  hexagonal  bipyramid^  of  the  first  order, 

1)    Since  these  are  really  double  pyramids,  the  term  bipyramid  is  employed. 


46 


HEXAGONAL    SYSTEM. 


Fig.  135. 


figure  135.  In  the  ideal  form,  the  faces  are 
all  equal,  isosceles  triangles.  The  symbols 
are  (aiOOa:  a:  c),  P,  jionf.1)  Because  the 
intercepts  along  the  c  and  two  secondary  axes 
are  taken  as  units,  such  bipyramids  are  also 
known  as  fundamental  or  unit  bipyramids, 
page  6. 

Planes  are,  however,  possible  which  cut 
the  two  secondary  axes  at  the  unit  distances, 
but  intercept  the  c  axis  at  the  distance  mc^ 
the  coefficient  m  being  some  rational  value 

smaller  or  greater  than  I,  see  page  6.  Such  bipyramids,  accord- 
ing as  m  is  greater  or  less  than  unity,  are  more  acute  or  obtuse  than 
the  fundamental  form.  They  are  termed  modified  hexagonal  bipyra- 
mids of  the  first  order.  Their  symbols  are  (a:  oca:  a:mc),  mP, 

\h  o  hl\,  where  m  =  y,  also  m  >  o  <  oc . 

The  principal  axis  passes  through  the  hexahedral  angles,  the 
secondary  axes  join  tetrahedral  angles,  while  the  intermediate  bisect 
the  horizontal  edges.  Hence  when  such  bipyramids  are  held  correctly, 
a  face  is  directed  towards  the  observer.  The  various  axes  of 
symmetry  are  located  by  means  of  the  above. 

2.  Hexagonal  bipyramid  of  the  second  order.  In  form,  this 
bipyramid  does  not  differ  from  the  preceding.  It  is,  however, 


Fig.  136. 


Fig.  137. 


i)    In  this  system  it  is  advantageous  to  employ  the  indices  as  modified  by  Bravais  (h  ikl)  rather 
than  those  of  Miller,  who  uses  but  three. 


DIHEXAGONAL    BIPYRAMIDAL    CLASS.  47 

to  be  distinguished  by  its  position  in  respect  to  the  secondary  axes. 
The  bipyramid  of  the  second  order  is  so  held  that  an  edge,  and  not  a 
face,  is  directed  towards  the  observer.  This  means  that  the  secondary 
axes  are  perpendicular  to  and  bisect  the  horizontal  edges  as  shown 
in  figure  136.  Figure  137  shows  the  cross  section  including  the  second- 
ary axes.  From  these  figures  it  is  obvious  that  each  face  cuts  one 
of  the  secondary  axes  at  a  unit  distance,  the  other  two  at  greater  but 
equal  distances.  For  example,  AB  cuts  az  at  the  unit  distance  OS, 
and  al  and  a2  at  greater  but  equal  distances  OM  and  ON,  respectively. 

The  following  considerations  will  determine  the  length  of  OM 
and  ON,  the  intercepts  on  al  and  «2,  in  terms  of  OS  =  i. 

As  already  indicated,  the  secondary  axes  are  perpendicular  to  the 
horizontal  edges,  hence  OS  and  ON  are  perpendicular  to  AB  and  BC, 
respectively.  Therefore,  in  the  right  triangleSpRB  and  NRB,  the 
side  RB  is  common  and  the  angles  OBR  and  NBR  are  equal. 1} 

Therefore,  OR  ==  RN.      But  OR  =  OS  ==  i. 

Hence,  ON  =  OR  +  RN  ==  2. 

In  the  same  manner  it  can  be  shown  that  the  intercept  on  a^ 
is  equal  to  that  along  «2,  that  is,  twice  the  unit  length.  The 
parametral  ratio  of  the  hexagonal  bipyramid  of  the  second  order, 
therefore,  is  (20:  2a:  a:  mc\  or  expressed  according  to  Naumann  and 

2/1 

Bravais,   mP2   and    \hh2h  l\,   where  -j- =  m.     -Figure   137  shows 

the  positions  of  the  bipyramids  of  both  orders  in  respect  to  the 
secondary  axes,  the  inner  outline  representing  that  of  first,  the  outer 
the  one  of  the  second  order. 


Fig.  138. 


i)     For,  angle  ABC  equals  120°,  angle  NBR  is  then  60°,  being  the  supplement  of  ABC.    But 
the  intermediate  axis  OZ  bisects  the  angle  ABC,  hence  angle  OBR  is  also  60°. 


48 


HEXAGONAL    SYSTEM. 


3.  Dihexagonal  bipyramid.  The  faces  of  this  form  cut  the 
three  secondary  axes  at  unequal  distances.  For  example,  in  figure 
138  the  face  represented  by  dB  cuts  the  a^  axis  at  A,  «2  at  C,  and 
as  at  B.  Assuming  the  shortest  of  these  intercepts  as  unity,  hence, 
OB  =  a  —  i,  we  at  once  see  that  one  of  these  axes  is  cut  at  a  unit's 
distance  from  O,  the  other  two,  however,  at  greater  distances.  If  we 
let  the  intercepts  ,OA  and  OC  be  represented  by  n(OB)  =  na,  and 
/>(OB)  =  pa,  respectively,  the  ratio  will  read 

na  :pa  :  a  :  me,  mPn,  \hikl\. 


In  this  ratio  p  — 


n 


Fig.  139. 


Twenty-four   planes   having  this  ratio 

are  possible  and  give  rise  to  the  form  called 
the  dihexag'onal  bipyramid,  figure  139. 
In  the  ideal  form  the  faces  are  equal, 
scalene  triangles,  cutting  in  twenty-four 
polar2),  a  and  b,  and  twelve  equal  basa!3) 
edges.  The  polar  edges  and  angles  are 
alternately  dissimilar.  This  is  shown  by 
figure  140,  where  the  heavy  inner  outline 
represents  the  form  of  the  first  order,  the 
outer  the  one  of  the  second,  and  the  inter- 
mediate outlines  the  dihexagonal  type  in 
respect  to  the  secondary  axes. 


z4ro, 


i)    From  the  above  discussion,  it  follows  that  in  figure  138 

OA:OC:  OB  =  «:/:!. 
Draw  XB  parallel  to  a2.  and  then 

OA:XA  =  OC:XB. 
But  the  triangle  OXB  is  equilateral, 

Hence,         XB  =  XO  =  OB  =  1. 

And  XA  =  OA  —  OX  =  OA—  1. 

Therefore,  OA  :  OA  —  1  =  OC  :  1. 

But  OA  =  n,  and  OC  =/, 

Hence,        w:n— !=/:!, 

°r  ^l  =  p- 

It  can  also  be  shown  that  the  algebraic  sum  of   the  Miller- Bra vais  indices  -{  hik  }•    is  equal  to 

h  +  i  +  ~k  =  o. 

For,  OA  =  -4-«  OC  =  4^»  and  OB 

n  t 

1 


-i->  page  12. 


Therefore,  -T-:   — =- r~  =  — 7-  :  — r-, 


J_.   J_          1. 

h   '      h    ~~    k 

Or,  k  =  h  + 1. 

Hence,  h  -\-  i  +  k  =  o. 

Those  joining  the  horizontal  and  principal  axes. 
These  lie  in  the  principal  plane  of  symmetry. 


DIHEXAGONAL    BIPYRAMIDAL    CLASS. 


49 


These  three  hexagonal  bipyramids  are  closely  related,  for,  if  we 
suppose  the  plane  represented  by  AB,  figure  140,  to  be  rotated  about 
the  point  B  so  that  the  intercept  along  a2  increases  in  length,  the  one 


Fig.  140. 

along  «j  decreases  until  it  equals  oB'  =  oB  =  i.  Then  the  plane  is 
parallel  to  a2  and  the  ratio  for  the  bipyramid  of  the  first  order  results. 
If,  however,  AB  is  rotated  so  that  the  intercept  along  a2  is  decreased 
in  length,  the  one  along  a^  increases  until  it  equals  oC  =  2oB'  =  2a. 
When  this  is  the  case,  the  intercept  on  «2  is  also  equal  to  2<7,  for  then 
the  plane  is  perpendicular  to  #3.  This  gives  rise  to  the  ratio  of  the 
bipyramid  of  the  second  order. 

That  the  bipyramids  of  the  first  and  second  orders  are  the  limit- 
ing forms  of  the  dihexagonal  bipyramid  is  also  shown  by  the  fact  that 

p  —  -      — .     For,  if  n  =  i,  it  follows  that/  =  00,  hence,  the  ratio 

of  the  form  of  the  first  order.  But,  when  n  —  2,  *p  =  2  also,  there- 
fore, the  ratio  for  the  second  order  results.  With  dihexagonal 
bipyramids  the  following  holds  good : 

n  >  i  <  2,  and/  >  2  <  00. 

The  dihexagonal  bipyramid  whose  polar  edges  and  angles  are  all 
equal  is  crystallographically  not  a  possible  form,  because  the  value  of 
n  would  then  be  ^(i  +  1/3)  =  1/2.  sin  75°  =  1.36603+,  which  of 
course  is  irrational.  It  also  follows  that  in  those  dihexagonal  bipyr- 
amids, where  the  value  of  n  is  less  than  1.36603  +  ,  for  instance, 
t  =  i. 20,  the  more  acute  pole  angles  indicate  the  location  of  the 
secondary  axes,  the  more  obtuse  that  of  the  intermediate,  and  vice 


50 


HEXAGONAL   SYSTEM. 


^r 

1  _J 

T\ 

i  I. 

i 

1 

l 

l 
1 

1 

1 
1 

1 

1 

v 

_  -J  r- 

-—  k 

Fig,  141. 


versa,   when  n  is  greater  than   1.36603  +  ,   for  example,    |-=i.6o. 
This  is  clearly  shown  by  figure  140. 

Hexagonal  prism  of  the  first  order.     This  form  is  easily  de- 
rived from  the  bipyramid  of  the  same  order  by  allowing  the  intercept 

along  the  c  axis  to  assume  its  maximum 
value,  infinity.  Then  the  twelve  planes  of 
the  bipyramid  are  reduced  to  six,  each 
plane  cutting  two  secondary  axes  at  the  unit 
distance  and  extending  parallel  to  the  c 
axis.  The  symbols  are  (a  :  oca  :  a:  oor), 
OoP,  |ioio|.  This  form  cannot  enclose 
space  and,  hence,  may  be  termed  an  open 
form,  page  6.  It  cannot  occur  indepen- 
dently and  is  always  to  be  observed  in 
combination,  figure  141.  The  secondary 
axes  join  opposite  edges,  i.  e.,  a  face  is 
directed  towards  the  observer  when  properly 
held. 


Hexagonal  prism  of  the  second  order. 

This  prism  bears  the  same  relation  to  the 
preceding  form  that  the  bipyramid  of  the 
second  order  does  to  the  one  of  the  first, 
page  46.  The  symbols  are  (2a:2a  :a  :  00  ^), 
00  P2,  jii2o[.  It  is,  hence,  an  open  form 
consisting  of  six  faces.  The  secondary  axes 
join  the  centers  of  opposite  faces,  hence, 
an  edge  is  directed  towards  the  observer, 
figure  142. 

Dihexagonal  prism.  This  form  may 
be  obtained  from  the  corresponding  bipyra- 
mid by  increasing  the  value  of  m  to  infinity, 
which  gives  (na  :  pa  :  a  :  ocr),  OoPw, 
\hiko\.  This  prism  consist  of  twelve 
faces  whose  alternate  intersection  angles  are 
dissimilar.  This  form,  figure  143,  is  closely 
related  to  the  corresponding  bipyramid  and, 
hence,  all  that  has  been  said  concerning 


Fig.  142. 


j 

j 

i 

i 



:C'.I!7 



'  ix 

i 

i 

i 

1  

-i-- 

i  

--  

i 

••  •  i 

_ 

Fig.  143. 


DIHEXAGONAL    BIPYRAMIDAL    CLASS. 


51 


the  dihexagonal  bipyramid,  page  48,  in  respect  to  the  location  of 
the  secondary  axes  and  its  limiting  forms  might  be  repeated  here, 
substituting,  of  course,  for  the  bipyramids  of  the  first  and  second 
orders  the  corresponding  prisms. 

7.  Hexagonal  basal  pinacoid.  The  faces  of  this  form  are 
parallel  to  the  principal  plane  of  symmetry  and  possess  the  following 
symbols  (cca  :  oca  :  0/oa:  c),  OP,  joooij.  It  is  evident  from  the 
presence  of  a  center  and  principal  plane  of  symmetry  that  two  such 
planes  are  possible.  This,  like  the  prisms,  is  an  open  form  and  must 
always  occur  in  combination.  Figure  141  shows  this  form  in  com- 
bination with  the  prism  of  the  first  order. 

These  are  the  seven  simple  forms  possible  in  this  system.  Their 
principal  features  may  be  summarized  as  follows: 


FORMS 

SYMBOLS 

Number  of  Faces  | 

Solid  Angles 

T  etrahedral 

Hexahedral 

Dodecahedral 

Weiss 

Naumann 

Miller- 
Biavais 

Unit  Bipyramid 
First  order 

a  :  <X  a  :  a  \c 

P 

\IOII\ 

12 

6 

2 



Modified 
Bipyramids 
First  order 

a:  oo  a  :  a  :  me 

mP 

\ho7i  l\ 

12 

6 

2 
2 



Bipyramids 
Second  order 

2a  :  2a  :  a  :  me 

mP2 

\hh~2lil\ 

12 

6 

— 

Dihexagonal 
Bipyramids 

na  :  fa  :  a  :  me 

mPn 

\hikl\ 

24 

6+6 

— 

2 

Prism 
First  order 

a\  oca:  a:  ccc 

ccP 

{  I  0  I  0  } 

6 

— 

— 

Prism 
Second  order 

2a  :  2a  :  a  :  oo  c 

ooP^ 

;  1  1  2  o  \ 

6 

— 

— 

Dihexagonal 
Prisms 

na  :  pa  :  a  :  oo  c 

ooPn 

\hilto\   12 

— 

— 

— 

Basal  Pinacoid         oca  :  oca  :  ooa  \c 

OP 

{  OOOI  \ 

2 

— 

— 

— 

52 


HEXAGONAL    SYSTEM. 


Relation  of  forms.  The  following  diagram,  similar  to  the  one 
for  the  cubic  system,  page  23,  expresses  very  clearly  the  relationship 
existing  between  the  various  simple  forms  . 


000  ;  oca  :  ooa  :  c 


a:  ooa  :  a  : 


a  :  ooa  :  a  :  ooc- 


na  :  pa  :  a  :  me 

I 

-na  :  pa  :a  :  ace- 


2a  :  2a  :  a  :  ooc 


Combinations.      The   following  figures  illustrate  some    of   the 
combinations  of  forms  of  this  class. 

Figure  144,^  =  P,   { icTi } ;  o  =  mP,   \holi l\. 
Figure  145,  p  =  P,  j  loir};  n  =    P2     {1121}. 


Fig.  144. 


Fig.  145. 


Fig.  146. 


Figure  146,  p  =  P,  { 101 1 } ;  p  =  |P  2,  12243^. 
Figure  147,  m  =  ooP,  \\oio]  ;p  =  P,  {ion}. 


DIHEXAGONAL    PYRAMIDAL    CLASS. 


53 


Figure  148,  m  =  ooP, 
Jioio};  p  =  P,  jioii};  u  = 
2P,  {2021} ;  c  =  OP,  { oooi  |; 
s  =  2P2;  jii2i};  and  v  = 

3  P|,  52131}.  This  combina- 
tion occurs  on  beryl,  Be3Al2 
(Si03), 

2.      DIHEXAGONAL    PYRAMIDAL 
CLASS.v 


Fig  147. 


Fig.  148. 


(Holohedrism  with  Hetnimorphism.} 

Symmetry.  The  forms  of  this  class  are  hemimorphic,  that  is, 
the  principal  plane  of  symmetry  disappears  and 
the  singular,  cy  axis  is  now  polar,  page  16. 
The  disappearance  of  the  principal  plane 
necessitates  also  the  loss  of  the  center  and  six 
binary  axes  of  symmetry.  Hence,  the  remain- 
ing elements  are  three  secondary  and  three 
intermediate  planes,  and  one  polar  axis  of 
hexagonal  symmetry  parallel  to  the  c  axis. 
Figure  149  shows  the  elements  of  symmetry 
of  this  class. 2) 

Forms.  The  forms  of  this  class  differ  from  those  of  the 
preceding  (holohedral)  in  that  the  faces  occurring  about  either 
pole3)  are  to  be  considered  as  belonging  to  independent  forms. 
Hence,  each  of  the  three  bipyramids  yields  two  new  forms,  called 
upper  and  lower  pyramids,  respectively.  For  example,  the  dihexa- 


Fig.  149. 


Fig.  150. 


Fig.  151. 


Fig.  152. 


1 )  Hemimorphic  group  of  Dana . 

2)  The  heavy  dotted  and  dashed  line  not  only  indicates  the  absence  of  the  principal  plane  but 
also  shows  that  hemimorpbism  is  effective.     Compare  pages  26,  32,  and  36. 

8)    The  ends  of  the  principal  or  c  axis  are  often  spoken  of  as  poles. 


54 


HEXAGONAL    SYSTEM. 


gonal  bipyramid  yields  the  upper  and  lower  dihexagonal  pyramids, 1} 
each  possessing  twelve  faces.  Figures  150,  151,  and  152  show  the 
upper  forms  of  the  three  bipyramids.  These  pyramids  are  open 
forms.  They  are  shown  in  combination  with  the  lower  basal 
pinacoid,  for  it  is  also  divided  into  an  tipper  and  lozuer  form  of  one 
face  each.  The  symbols  of  these  forms  are  the  same  as  those  used 
for  the  holohedral,  except  that  the  end  of  the  c  axis  is  indicated 
about  which  they  occur.  Since,  however,  the  development  of  the 
prisms  is  the  same  about  both  poles,  no  new  forms  result  from  them. 
The  principal  features  and  symbols  of  the  forms  of  this  class  are 
given  in  the  following  table: 


FORMS 

SYMBOLS 

Number  of  Faces 

Solid 
Angles 

Number  of  Edges 

Hexahedral 

Dodecahedral 

Weiss 

Naumann 

Miller- 

Bravais 

Upper  and  Lower 
Pyramids 
First  Order 

a  :  OO  (i  :  a  :  me 

f?" 

K-' 

\  ho  hi] 
{  ho7i~r\ 

1 

0 

F 

1 
1 

6 
6 

2 



Upper   and  Lower 
Pyramids 
Second  Order 

2ct  :  2(i  :  a  :  me 

1L     I 

\mP'  u 

\hhTh  l\ 
\hh2h"l\ 

\  * 

j  mP2 

I      2 

2                    U^ 

Upper  and   Lower 
Dihexagonal 
Pyramids 

na  :  pa  :  a  :  me 

(mPn 

-  u 

\     2 
\  mPn 

\kikl  \ 

\hil7\ 

i» 

1 

6+t> 

2 

I      2 

Prism 
First  Order 

a  :  OCrt  :  a  :  O0£ 

OOP 

OOP? 

j  IOIO  \ 

6 

1 

Same  as 
in  holo- 
hedral 
class 

j 

Prism 
Second  Order 

20,  :  20.  :  a  :  OOc 

{1120} 

6- 

Dihexagonal 
Prism 

na  :  pa  :  a  :  OO  c 

mPn 

\hi~k  o\ 

12 

F 

Upper  and   Lower 
Basal  Pinacoids 

O0«  :  OCa  :  OC«  :  c 

(°Pu   I 

1  0001  j- 

{0001  j 

— 

— 

— 

2                  ",t 

!_«,/ 

1)     The  term  pyramid  in  itself  suggests  hemimorphism,  since  it  is  not  a  doubly  terminated  form 
as  is  the  bipyramid. 


HEXAGONAL    HEM1HEDRISMS. 


55 


Combinations. 

OP 
Figure  153,  <:  =     —  u, 

P 

I oooi  j ;   o  =    -  u  &/, 

jiouj  &  {lonf ;  i  = 

2p  ip 

—  u  &  /,  52021!  &  52021};  v  =  — /,  {2023 j, 

ip 
and  /*=•—/,  |ioi2c ;  observed  on  lodyrite,  Agl. 


Fig.  154. 


Figure  154,  p  -      -  w,    {ioii[;    w  =  ooP,  {1010},  observed  on 
Zincite,  ZnO. 

HEXAGONAL   HEMIHEDRISMS. 

In  the  hexagonal  system  four  types  of  hemihedrism,  as  illustrated 
by  figures  155  to  158,  are  possible. 


Fig.  155. 


Fig.  156. 


Fig.  157. 


Fig.  158. 


a)  Trigonal   hemihedrism.      The    three   secondary  planes  of 
symmetry  divide  space  into  six  equal  sections. x)    All  faces  in  alternate 
sections  are  extended,  the  others  suppressed,  figure  155. 

b)  Rhombohedral   hemihedrism.     The  three  secondary  planes 
together  with  the  principal  plane  divide  space  into  twelve  sections, 
called    dodecants,    page    45.      Faces    in    alternate    dodecants    are 
subject  to  extension,  figure  156. 

c)  Pyramidal   hemihedrism.      By  means  of  the  three  second- 
ary and  three  intermediate  planes  twelve  sections,  figure  157,  result. 
Faces  in  alternate  sections  are  suppressed. 

i)  Liebisch,  however,  refers  to  the  sections  derived  by  the  intersection  of  the  intermediate 
instead  of  the  secondary  planes.  The  forms,  which  result,  are  in  both  cases  identical.  Their  positions 
in  respect  to  the  secondary  axes  differ  by  an  angle  of  80°.  Compare  Liebisch,  Grundriss  der 
Physikalischen  Krystallographie,  1896,  112. 


56  HEXAGONAL   SYSTEM. 

d)  Trapezohedral  hemihedrism.  All  planes  of  symmetry 
possible  in  this  system  divide  space  into  twenty-four  sections.  The 
extension  of  faces  occurs  in  alternate  sections  of  this  character, 
figure  158. 

Of  these  hemihedrisms,  the  rhombohedral  and  pyramidal  types 
are  the  most  important.  No  representative  of  the  trigonal 
hemihedrism  has  yet  been  observed. 

3.    DITRIOONAL  BIPYRAMIDAL   CLASS. 

(  Trigonal  Hemihedrism.} 

Symmetry.  From  figure  155  showing  this  method  of  hemi- 
hedrism it  is  obvious  that  the  secondary 
planes  of  symmetry  are  lost.  With  them  the 
center  and  the  axis  of  hexagonal  symmetry 
also  disappear.  The  elements  of  this  class 
are,  hence,  one  principal  and  three  interme- 
diate planes,  three  binary  axes  parallel  to  the 
intermediate,  and  an  axis  of  trigonal  symmetry 
parallel  to  the  c  axis.  The  binary  axes  are 
polar.  Figure  159  not  only  shows  these 
Fig.  159.  elements  but  the  application  of  the  hemihe- 

drism as  well. 

Trigonal  bipyramids.  From  the  hexagonal  bipyramid  of  the 
first  order  two  correlated,  congruent  forms,  each  bounded  by  six 
equal  isosceles  triangles,  result.  These  forms  and  their  position  in 
respect  to  the  secondary  axes  are  shown  by  figures  160-165.  They 

,  a:  00  a:  a:  me 
are  the  trigonal  bipyramids.     The  symbols  are  j 


mP  — 

hohl\i  figure  162;  —     —  ,  \ohhl\,    figure  160. 


-      2  2 

The  axis  joining  the  trihedral  angles  is  of  one  of  trigonal 
symmetry,  those  passing  through  the  centers  of  the  horizontal  edges 
to  the  opposite  tetrahedral  angles  are  of  binary  symmetry.  These 
are  parallel  to  the  intermediate  axes. 

Trigonal  prisms.  The  corresponding  hexagonal  prism  yields 
two  trigonal  prisms,  1}  each  bounded  by  three  faces,  as  is  shown  by 
figures  163  to  168. 


Being  open  forms,  they  are  shown  in  combination  with  the  basal  pinacoid. 


DITRIGONAL    BIPYRAMIDAL    CLASS. 


57 


Fig.  160. 


Fig.  163. 


Fig.  166. 
The    symbols   are : 

OOP 


Fig.  161. 


Fig.  162. 


Fig.  165. 


Fig.  167. 


Fig.  168. 


,  a:  oca:  a:  oo^l  ooP  - 

+    I-  I!  +         -  ,  \hoho\, 


figure  1 68;  — 


ohho\,  figure  166. 


The  trigonal  axis  is  parallel  to  the  intersection  lines  of  the  prism 
faces,  while  those  of  binary  symmetry  pass  from  the  centers  of  the 
faces  to  those  of  the  opposite  edges,  figures  166  and  168. 

Ditrigonal  bipyramids.  Every  dihexagonal  bipyramid,  when 
subjected  to  the  trigonal  hemihedrism,  gives  rise  to  two  correlated 
and  congruent  forms,  bounded  by  twelve  scalene  triangles,  which  are 


58 


HEXAGONAL    SYSTEM. 


known  as  the  ditrig'onal  bipyramids.      The  derivation  and  position 
of  these  forms  are  illustrated  by  figures  169  to  174. 

_,  \na  :  pa  :  a  :  me]          mPn 

The  symbols  are:  +_  -y-  ,  -| — ,   \hikl\,   figure 

in  P  n    .      —  . 
171; ,  \ihkl\,  figure  169. 

The  trigonal  axis  joins  the  hexahedral  angles,  those  of  binary 
symmetry  extend  from  an  obtuse  to  an  opposite  more  acute  tetrahe- 
dral  angle. 


Fig.  172. 


I 
I 

j_- 


Fig.  175. 


Fig.  176. 


Fig.  177. 


DITRIGONAL    BIPYRAMIDAL    CLASS. 


59 


Ditrigonal  prisms.  The  dihexagonal  prism  yields  two  ditri- 
gonal  prisms,  as  shown  by  figures  172  to  177.  The  symbols  are 

[na:pa\a\  oo^l          ocP;;                                              mPn 
±  2 j  ,  +  -^— ,  \hiko\,  figure  177; — ,   \ihko\ 

figure  175. 

The  trigonal  axis  is  parallel  to  the  intersection  lines  of  the  prism 
faces,  those  of  binary  symmetry  are  parallel  to  the  intermediate  axes. 

Other  forms.  The  bipyramids  and  prism  of  the  second  order  as 
also  the  basal  pinacoid  are  not  changed  geometrically  by  the  trigonal 
hemihedrism,  for  none  of  their  faces  lie  wholly  within  the  sections 
formed  by  the  secondary  axes.  Compare  figures  159  and  164.  These 
forms  are,  therefore,  apparently  holohedral. 

No  representative  of  this  class  has  yet  been  discovered. 

The  chief  characteristic  of  the  forms  may  be  tabulated  as  follows: 


FORMS 

SYMBOLS 

1  Number 
of  Faces 

Solid  Angles 

Trigonal 

Tetrahe- 
dral 

Hexahe- 
dral 

Weiss 

Naumann 

Miller- 
Bravais 

Trigonal 
Bipyramids 
First  order 

_L_  (  a  :OCa:a  :mc~\ 

f+r 

ntP 

2 

{ho7ik} 

{ohlik} 

[    6 
j 

2 

3 

L(               2                } 

Hexagonal 
Bipyramids 
Second  order 

2a\2a\a\  me 

mP2 

{hh~2hl} 

Apparently  holohedral 

Ditrigonal 
Bipyramids 

,    [na  \pa\a  :  mc~\ 

f  +  *5! 

2 
}       mPn 

{  hiKl] 
{i  h  ic/  } 

1 

[12 

3+3 

2 

-LI      2       ] 

(               2. 

Trigonal  Prisms 
First  order 

,    fa:aca:a:OC^ 

r        rjn  p 
1-4- 

(hoTio  } 
{o  hH.  0} 

1. 

j 

• 

— 

— 

2 

OOP 

-U           2            J 

(        -    2 

Hexagonal  Prism 
Second  order 

2  a  :  20,  :  a  :  00  c 

OCP2 

\    I  I  20\ 

Apparently  holohedral 

Ditrigonal  Prisms 

_L_  [na  \pa:  OCa:c^ 

r   ,J~Pn 

\hfko  \ 
\ih~ko  \ 

j. 

— 

\    '      ^ 
CfOPn 

±1                 2                J 

{            2 

Basal  Pinacoid 

OCa  :  OOa  :CCa  \c 

OP 

{o  o  o  i  \ 

Apparently  holohedral 

60 


HEXAGONAL    SYSTEM. 


4.     D1TRIQONAL    SCALENOHEDRAL    CLASS.*) 


Fig.  178. 


(Rhombohedral  ffemihedrism.} 

Symmetry.  The  principal  and  second- 
ary planes  of  symmetry  together  with  the 
hexagonal  axis  are  lost.  The  following 
elements  are  present:  three  intermediate 
planes,  three  axes  of  binary  and  one  of 
trigonal  symmetry,  also  the  center.  Figure 
178  shows  these  elements  and  the  applica- 
tion of  the  rhombohedral  hemihedrism, 
page  55. 


Rhombohedrons.  From  the  hexagonal  bipyramid  of  the  first 
order  two  new  congruent  forms  are  the  result  of  this  type  of  hemihe- 
drism, figures  181  to  183.  In  the  ideal  development  each  of  these 
forms  is  bounded  by  six  equal  rhombs  and  are  called  positive  and 

negative  rhombohedrons.  The  lateral 
edges  form  a  zigzag  line  about  the  torm. 
The  six  polar  edges  form  two  equal  trihe- 
dral angles,  bounded  by  equal  edges. 
These  may  be  larger  or  smaller  than  the 
other  trihedral  angles,  according  to  the 
value  of  a:c.2^ 


i)     Dana  terms  this 
class    the    rhombohedral 


Fig.    180. 


Fig.  179. 


2)  The  cube,  when 
held  so  that  one  of  its  axes 
of  trigonal  symmetry,  page 
IS,  is  vertical,  may  be  con- 
sidered as  a  rhombohe- 
dron  whose  edges  and 
angles  are  equal.  "The 
ratio,  a  :  c,  in  this  case 

would  be  1  :  T/1.5  =  1  :  1.2247+.  Those  rhombohedrons,  there- 
fore, whose  c  axes  have  a  greater  value  than  1.2247+  have 
pole  angles  less  than  90°.  When,  however,  the  value  is  less  than 
1.2217+,  the  pole  angles  are  then  greater  than  90°  and,  hence, 
such  rhombohedrons  may  be  spoken  of  as  acute  and  obtuse, 
respectively,  figures  179  and  180. 


DITRIGONAL    SCALENOHEDRAL    CLASS. 


61 


Fig.  181. 


Fig.  182. 


Fig.  183. 


~u  (a:oca:a:mc]  mP  T 

The  symbols  are  ±_  |  ;  +  —,K\hohl\,  figure 

183; — ,    K\ohhl\,  figure  181. 

The  principal  crystallographic  axis  passes  through  the  two  equal 
trihedral  angles,  the  secondary  axes  bisect  opposite  lateral  edges. 
These  axes  indicate  the  directions  of  those  of  trigonal  and  binary 
symmetry,  respectively. 

Scalenohedrons.  Every  dihexagonal  bipyramid  gives  rise  to  two 
congruent  forms,  bounded  by  twelve  similar  scalene  triangles,  called 
scalenohedrons,  of  which  one  is  positive  and  the  other  negative, 
figures  184  to  1 86.  Each  possesses  six  obtuse  and  six  more  acute 
polar  edges,  also  six  zigzag  lateral  edges.  As  is  the  case  with  the 
rhombohedrons,  obtuse  and  acute  scalenohedrons  are  possible, 
depending  upon  the  value  of  a  :  c. 


Fig.  184. 


Fig.  185. 


Fig.  186, 


62 


HEXAGONAL    SYSTEM. 


._,                 .                  \na  :pa  :  a\mc\ 
The  symbols  are :  -f-  


m  Pn 


,  K\hikl\,  figure 


186;  - 


,   figure  184. 


Scalenohedrons  with  twelve  equal  polar  edges  are  crystal- 
lographically  impossible,  see  page  49. 

The  axis  of  trigonal  symmetry  passes  through  the  two  hexahe- 
dral  angles,  while  those  of  binary  symmetry  bisect  the  lateral  edges. 

The  other  holohedral  forms  remain  unchanged  by  the  application 
of  the  rhombohedral  hemihedrism,  compare  figures  178,  182,  and  185. 

Abbreviated  Symbols  of  Naumann.  In  order  to  more  easily 
express  some  of  the  interesting  relationships  existing  between  forms 

of  this    class,    which   are    quite    common, 

Naumann    substituted  for 


P 

—  and-f 

the  symbols  of  the  rhombohedrons  used 
above,  +R  and  4^  mR,  respectively.  For 
every  rhombohedron  there  exists  a  series  of 
scalenohedrons,  whose  lateral  edges  coin- 
cide with  those  of  the  rhombohedron,  as 
shown  in  figure  187.  The  inscribed  rhom- 
bohedron is  known  as  "the  rhombohe- 
dron of  the  middle  edges.  "  The  scalen- 
ohedrons may,  therefore,  be  indicated  in 
general  by  mRn,  mR  representing  the 
"rhombohedron  of  the  middle  edges.  "  In 
mRn,  n  has  reference  to  the  value  of  the  c 
axis1}  of  the  scalenohedron  in  respect  to  that 
of  the  rhombohedron  mR.  For  example, 
in  figure  187,  the  length  of  the  c  axis  of  the 
scalenohedron  is  three  times  that  of  the 
inscribed  positive  unit  rhombohedron.  The 

symbol  for  this  scalenohedron  is,  hence,  according  to  Naumann  -{-R3. 
The  prism  of  the  first  order  OoP,  also  the  basal  pinacoid  OP, 

are  often  written   QoR  and  OR,  respectively.     The  symbols  for  the 

other  forms  mP2,  QoP2,  ooPw,  remain  unchanged. 


Fig.  187. 


And  not  to  the  secondary  axes,  as  is  usually  the  case  in  the  regular  Naumann  symbols. 


DITRIGONAL    SCALENOHEDRAL    CLASS. 


To   transform  the  full   into  the  abbreviated  symbols  and  vice 
versa,  the  following  formulae  are  useful: 


2. 


n 


_»_^  4PI  = 

2  —  n  2 


mnP 


2H 


— .      For  example,  2R2  =  — -. 


3.  *\hikl\    = 

4.  mRn    =  K 


R 


Here,  K 


=  2R2. 


I         v  2k— ti 

'.    —  (n—  i).        -  (n  -\-  i).      ~\-      For 

instance,    2R2  =    ^4131}. 

The  principal  features  of  the  forms  of  this  class  are  given  in  the 
following  table: 


FORMS 

SYMBOLS 

* 

Number  of  Faces 

So'td  Angles 

"3 

T3 

J^ 

H 

Tttrahedial 

1  Hexahedral 

Weiss 

Naumann 

Miller- 
Bravais 

Rhombohe- 
drons 

f  a  :  cr.  a  :  a  •  :  me  1 

j   +mR 

I    —mR 

*\hohl\ 

\6 

J 

-?    '  (^ 

'  I 
-I          '*            J 

K\o/ihl\ 

2    |  O 

Hexagonal 
Bipyramid 
Second  order 

2a:  2a:  a  :  me 

mP2 

K\/l/l27ll\ 

Apparently 
holohedral 

Scalenohedrons 

,   \nd\-pa\a\mc\ 

'                           1 

\-  +  mRn 
|  —mRn 

*\hikl\ 
*\ihkl\ 

} 

•12 

6 

2 

-I                 2                j 

Hexagonal 
Prism 
First  order 

a  :  oca  :  a  :  00  c 

GOT? 

\-\hoho\ 

1 

Ap 

j 

parei 
holol 

Itl) 

tied 

T 

ral 

Hexagonal 
Prism 
Second  order 

20,  :  2a  :  a  :  CCc 

CCP2 

K  \II20\ 

Dihexagonal 
Prism 

na  :  pa  :  a  :  GO  c 

tePn 

K  {  hiko  \ 

Basal  Pinacoid 

ooa  :  cca  :  ooa  :  c 

OR 

K  {  OOOI  \ 

64 


HEXAGONAL    SYSTEM. 


Combinations.  Many  of  the  more  common  minerals  crystallize 
in  this  class,  for  example,  Calcite,  CaCO3 ;  Hematite,  Fe2O3 ;  Corun- 
dum, A12O3 ;  and  Siderite,  FeCO3. 


Fig.  188. 


Fig.  189. 


Fi-   190. 


Fig.  191. 


Figure  188.  r  =  R,  K  j  101 1  \ ;    c  =  OR,  K\OOOI}._    Calcite. 

Figure  189.  e  =  — ^R,  K{oii.2} ;  m  =  OoR,  K\  ioTp}.    Calcite. 

Figure  190.  e  =  — iR,  K{oii2(;   a  =  OoPa,  K{  1 120}.  Calcite. 

Figure  191.  r  =  R,  K{IOII};  /=  — 2R,  K{o22i\.     Calcite. 


Fig.  192. 


Fig.  193. 


Fig.  194. 


Figure  192.     v  =  RS,    KJ2I3I};   r  =  R,  KJIOII}.      Calcite. 
Figure  193.     r  —  R,    KJIOII};   v  =  R3,  ^{21^1  \ ._  Calcite. 
Figure  194.     c  =  OR,   K{OOOI  } ;    m  =  OoR,  K{  loiol ;  s  =  R$ 
;  0  =  QC-P2,  K-Jii2o}.     Calcite. 


Fig.  195. 


Fig.  196. 


Fig.  197. 


HEXAGONAL    BIPYRAMIDAL    CLASS. 


65 


Fig.  198. 


Figure  195.  d  =  OR,  KJOOOI|;  r  =  R, 
K{IOII|.  Corundum. 

Figure  196.  d  —  OR,  K|OOOI|;  r  —  R, 
KJioiif ;  n  =  |P2,  KJ2243J;  I  =  OoP2,  K{  1 120}. 
Corundum. 

Figure  197.  r  =  R,  KJIOII};  r'  =  £R, 
K\ioi4\',  p  =  |P2,  K 1 2243}.  Hematite. 

Figure  198.  r  =  R,  KJIOII j;  4r  =  4R, 
*J404il;  w  =  R5,  «S325i};  *  =  R3»  KJ2i3if; 
g-  =  ooR,  K{  loioj.  Calcite. 

6.     HEXAGONAL    BIPYRAMIDAL    CLASS.*) 

(Pyramidal  Hemihedrism.) 

Symmetry.  The  secondary  and  intermediate  planes  and  the 
six  axes  of  binary  symmetry  are  lost.  The  remaining  elements  are 
the  principal  plane,  the  hexagonal  axis  and  the  center  of  symmetry, 
figure  203,  page  66. 

Hexagonal  bipyramids  of  the  third  order.  Figure  200  illus- 
trates the  pyramidal  method  of  extension  and  suppression  of  faces 
on  the  dihexagonal  bipyramid.  It  is  obvious  that  this  bipyramid 
yields  two  new  forms,  each  bounded  by  twelve  equal  isosceles  tri- 
angles. They  are  termed  positive  (figure  201)  and  negative*^  (figure 
199)  hexagonal  bipyramids  of  the  third  order.  In  form  these 
bipyramids  do  not  differ  from  those  of  the  first  and  second  orders. 
Figures  202  and  204  show  the  positions  of  these  bipyramids  in 
respect  to  the  secondary  axes.  The  inner  and  outer  light  outlines 
represent  the  horizontal  cross-sections  of  the  bipyramids  of  the  first 
and  second  orders,  respectively,  while  the  heavy  outlines  show  the 
cross-sections  of  those  of  the  third  order.  These  occupy  an  inter- 
mediate position. 


,     {na\pa\a\  me]       ,    mPn 

I          ~ '      i~    


The  symbols  are:     + 

I  4 

iare2Oi;  —        — ,  ir\ihkl\    figure  199. 


TT\/likl\       fig- 


I)"   Called  by  Dana  the  pyramidal group, 

2)    The  terms  right  and  left  are  sometimes  used. 


66 


HEXAGONAL   SYSTEM. 


The  axis  of  hexagonal  symmetry  passes  through  the  hexahedral 
angles.  The  position  of  the  secondary  crystallographic  axes  is  shown 
by  figures  202  and  204.  These  do  not  join  the  tetrahedral  angles 
or  the  centers  of  the  basal  edges,  but  some  point  between  them, 
which  is  dependent  upon  the  value  of  n.  Compare  figures  135  to  136. 


Fig.  199. 


Fig.  202. 


Fig.  200. 


Fig.  203. 


Fig.  201. 


Fig.  204. 


Fig.  205. 


Fig.  206. 


Fig.  207. 


HEXAGONAL    BIPYRAMIDAL    CLASS. 


67 


Hexagonal  prisms  of  the  third  order.  From  the  dihexagonal 
prism  two  congruent  forms  consisting  of  six  planes,  the  positive 
(figure  207)  and  negative  (figure  205)  hexagonal  prisms  of  the 
third  order  are  derived.  Figures  202  and  204  show  these  forms  and 
their  relation  to  the  other  hexagonal  prisms.  Their  position  in 
regard  to  the  secondary  axes  is  the  same  as  for  the  bipyramids  of  this 
order. 


_, 

The  symbols  are:     ±_ 


na  :pa  :  a  :  ccc 


;H 


ooPn 


\hiko 


figure   207;  —  —  ,  Tt\ihko\  figure  205. 


The  axis  of  hexagonal  symmetry  extends  parallel  to  the  edges. 

The  other  holohedral  forms  are  unaltered  by  the  application  of 
the  pyramidal  hemihedrism,  since  the  suppression  effects  only  one  half 
of  each  face.  Compare  figures  200,  203  and  206.  They  are,  hence, 
apparently  holohedral. 

The  principal  features  of  this  class  have  been  summarized  in  the 
following  table: 


FORMS 

SYMBOLS 

"o 

|i- 

§^ 

Solid 
'Angles 

u 

EC 

aJ-o 
H 

arenl 
lohe 

i 

la 

HTJ 

as 
iy 

dral 

Weiss 

Naumann 

Miller- 
Bravais 

Hexagonal 
Bipyramids 
First  order 

a  :  O0a:a:mc 

mP 

7f\hoHl\ 

i 

App 
ho 

J 

Hexagonal 
Bipyramids 
Second  order 

20,  :  20,  :  a  :  me 

mP2 

TT\hh2hl} 

Hexagonal 
Bipyramids 
Third  order 

,   f  na  :  pa  :  a  :  mc\ 

f         mPn 

TT\    hill   \ 

ir\i  h~k  l\ 

u 

J 

6 

2 

\              2 
mPn 

1l       -       J 

('         * 

Hexagonal  Prism 
First  order 

a  :  oca  :  a  :  OOr 

OOP 

TT  |  k  O  h  0| 

App 
1      ho 

J 

arent 
lohe< 

iy 

iral 

Hexagonal  Prism 
Second  order 

20.  :  20,  :  a  :  OO  c 

OOP2 

7TJ7  /  20   \ 

Hexagonal  Prisms 
Third  order 

_j_  (  na  :  pa  :  a  :  OOc  ] 

f         CCPn 

ir\hiJo\ 
ir\ihk~o  \ 

r 

— 

— 

+       - 
OO  Pn 

-{               '              J 

\ 

Basal  Pinacoid 

CCa  :  OOa  :  OOa  :  c 

OP 

ir\o  o  01  \ 

Apparently 
holohedral 

68 


HEXAGONAL   SYSTEM. 


Fig.  208. 


Combinations.     Figure  208.      m   =  oop? 
;   c  =  OP,  TrjoooiS;   x  --  P,  TrjiouJ; 


5  =  2P2,  irjll2l|;  U    =    +  ,  ir  j  I  231}.    This 

combination    has    been    observed    on    apatite, 
Ca5Cl(P04), 


HBXAQONAL   TRAPBZOHEDRAL   CLASSY) 

(  Trapezohedral  Hemihedrism. ) 

Symmetry.  The  center  and  all  planes 
of  symmetry  disappear.  Hence,  the  ele- 
ments of  this  class  are  the  hexagonal  axis 
and  six  binary  axes  of  symmetry,  figure  209. 

Hexagonal  Trapezohedrons.  From 
figure  211  it  is  evident  that  by  means  of  this 
hemihedrism,  the  dihexagonal  bipyramid 
yields  two  correlated  forms  bounded  by 
similar  trapeziums.  The  forms  are  enan- 
tiomorphous  and  designated  as  the  right 
and  left  hexagonal  trapezohedrons. 


.  209. 


Fig.  210. 

Their  symbols  are:     r,  I 

_  in  Pn 
figure  212;   l——,T\kihl\  figure  210. 


Fig.  211. 

f  na  :  fia  :  a  :  me] 
•*       . 


Fig.  212. 


:,  r\hikl\ 


1)     Trapezohedral  group  of  Dana. 


HEXAGONAL    TRAPEZOHEDRAL    CLASS. 


69 


On  the  right  trapezohedrons  the  six  longer  basal  edges  are 
inclined  to  the  left  of  the  observer  and,  vice  versa,  to  the  right  in  the 
case  of  the  left  form,  compare  figures  212  and  210. 

The  axis  of  hexagonal  symmetry  joins  the  hexahedral  angles. 
The  secondary  crystallographic  axes  connect  the  centers  of  the  longer 
basal  edges,  the  intermediate  those  of  the  shorter.  These  axes 
possess  binary  symmetry. 

Since  this  hemihedrism  calls  for  the  suppression  of  only  portions 
of  each  of  the  faces  of  the  other  holohedrons,  no  new  forms  can  be 
derived  from  them.  They  are  apparently  holohedral. 

The  chief  features  of  the  forms  of  this  class  are  given  in  the 
following  table: 


FORMS 

SYMBOLS 

Number  of  Faces 

Sohd 
Angles 

Trihedral 

Hexahedral 

Weiss 

Naumann 

Miller-Bravais 

Hexagonal 
Bipyramids 
First  order 

a  :  oca  :  a  :  me 

mP 

r\hohl\ 

Appar- 
ently 
holohe- 
dral 

Hexagonal 
Bipyramids 
Second  order 

2(i  :  2a  :  a  :  me 

mP2 

T\hh2~hl\ 

Hexagonal 
Trapezohedrons 

1  (  na:  pa:  a:  me] 

f    mPn 

r\hikl\ 
T\kihl\ 

} 

12 

12 

2 

r  ; 

mPn 

'"[      *      J 

I/        2 

Hexagonal  Prism 
First  order 

a  :  oo  a  :  a  :  oc  c 

OOP 

T\hoho\ 

1 

AF 

ho 

j 

>pa 
ent 
loh 
di 

r- 

iy 

e- 
•al 

Hexagonal  Prism 
Second  order 

2a  :  2a  :  a  :  occ 

00  P2 

T\II20\ 

Dihexagonal 
Prism 

na  :  pa  :  a  :  occ 

oo  Pn 

r\hiko\ 

Basal  Pinacoid 

oca  :  ooa  :  ooa  :  c 

OP 

r  \OOOl} 

70 


HEXAGONAL   SYSTEM. 


No  crystals  showing  the  occurrence  of  the  hexagonal  trapezo- 
hedrons  have  as  yet  been  discovered.  The  double  salt,  barium  stibio- 
tartrate  and  potassium  nitrate,  Ba(SbO)2  (C4H4O6)2  +  KN  O3,  and  the 
corresponding  lead  salt  are  assigned  to  this  class.  The  crystals  are 
apparently  holohedral,  the  etch  figures,  however,  reveal  the  lower 
grade  of  symmetry. 

7.    DITRIQONAL  PYRAMIDAL   CLASS.V 

(  Trigonal  Hemihedrism  with  Hemimorphism. ) 

Symmetry.  If  the  forms  of  the  ditri- 
gonal  bipyramidal  class,  page  56,  become 
hemimorphic,  the  principal  plane  of  symmetry 
and,  of  course,  the  axes  of  binary  symmetry 
also  are  lost.  Three  intermediate  planes  and 
a  polar  axis  of  trigonal  symmetry  parallel  to 
the  c  axis  are  the  only  elements  left.  These 
symmetry  relations  are  shown  in  figure  213. 
The  forms  of  this  class  may,  moreover,  be  considered  also  as 
derived  from  those  of  the  ditrigonal  scalenohedral  class,  that  is,  from 
those  showing  rhombohedral  hemihedrism,  page  60.  A  comparison 
of  figures  159  and  178  reveals  the  fact  that  when  hemimorphism 
becomes  effective  the  resulting  forms  must  in  both  cases  be  the  same. 
Therefore,  this  class  is  often  known  as  the  rhombohedral  hemi- 
morphic class. 

Trigonal  pyramid  of  the  first  order.  The  trigonal  bipyramid, 
page  56,  the  result  of  the  application  of  the  trigonal  hemihedrism  to 
the  hexagonal  bipyramid  of  the  first  order,  yields  upon  the  addition 
of  hemimorphism  upper  and  lower  forms,  consisting  of  but  three 
faces  each.  Since  there  are  two  trigonal  bipyramids,  one  positive 
and  the  other  negative,  it  follows  that  each  hexagonal  bipyramid  of 
the  first  order  now  yields  four  forms. 2)  Being  singly  terminated,  they 
are  termed  trigonal  pyramids. 3) 


Fig.  213. 


!)     Called  by  Dana  the  rhombohedral  hemimorphic  class. 

2)  The  forms  are,  hence,  sometimes  said  to  be  the  result  of  tetartohedrism,  namely  the  ditri- 
gonal Pyramidal  tetartohedrism . 

3)  Compare  footnote  on  page  51.     The  letters  u  and  /  designate  the   upper  and  lower  forms, 
respectively. 


DITRIGONAL    PYRAMIDAL   CLASS.  71 

Their  symbols  are: 

f  a  :  oo a  :  a  :  mc\  m P 

Positive  upper,  +  — —  I  u,      ~u,   \hohl\. 


f «  :  oo  a  :  a  :  me]  .        m  P       .        7- 
Positive  lower,   -H  •/,  +  •-/,  {Ao A/}. 

4  4 

fa  :  oca  :  a  :  me]  mP 

Negative  upper,  -  — — -  \u,  -    —  u,   \o/i/il\. 

[a  :  oca  :  a  :  me]  mP 

Negative  lower,  -  — —  /,   -    — -  /,   \ohhl\. 

Hexagonal  pyramids  of  the  second  order.  The  hexagonal 
bipyramid  of  this  order  remains  unaffected  by  the  trigonal  hemihe- 
drism, page  59,  but  now  yields  an  upper  and  lozuer  form,  each 
consisting  of  six  faces,  termed  hexagonal  pyramids  of  the  second 
order. 

( 2a  :  2a  :  a  :  me^  mP2 

The  symbols  are: \u,   1;     —^-U1{hh2hl\, 

?,  \hh~2lil\. 


Ditrigonal  pyramids.  The  dihexagonal  bipyramid  by  means 
of  the  trigonal  hemihedrism  yields  the  positive  and  negative  ditrigonal 
bipyramids,  page  57.  Each  of  these  bipyramids  now  gives  rise  to 
an  upper  and  lower  form,  termed  ditrigonal  pyramids. 1}  These, 
like  the  preceding  pyramids  of  this  class,  are  open  forms.  They 
consist  of  six  faces. 

The  symbols  are: 

[na  :  pa  :  a  :  me]             mPn 
Positive  upper,  -  •  \  u,    -\ u,  \hik l\. 

I  4  J  4 


Positive  lower, 


na:Pa:a:me]   ^  +  mPn  ^          -- 


!)  Since,  as  said  on  page  70,  these  forms  may  be  considered  as  derived  from  those  showing 
rhombohedral  hemihedrism,  the  terms  rhombohedron-  and  scalenohedron-like  faces  are  sometimes  used 
for  the  trigonal  and  ditrigonal  pyramids,  respectively,  the  forms  being  assumed  to  be  half  of  the 
rhombohedron  and  scalenohedron. 


72  HEXAGONAL   SYSTEM. 

\na  :  pa  :a  :  me}  mPn  ,7-75 

Negative  upper,  —    -  I  «,  —  --  u,  \thkl\. 

I  4  J  4 

^  na  :  pa  :  a  :  mc\  mPn    .    ,.,--=, 

Negative  lower,  —  I  -  /,  —  -       -  /,  {thkl\. 

I  4  J  4 

Trigonal  prisms.  Hemimorphism  does  not  affect,  morphologi- 
cally, the  trigonal  prisms,  the  hemihedrons  of  the  hexagonal  prism 
of  the  first  order,  page  56,  for  each  face  belongs  alike  to  the  upper 
and  lower  poles. 

a:  aoa:a:  ccc  OoP 


The    symbols    are  :     + 


J   -     —      »     \nono\\ 


~ 


Hexagonal  prism  of  the  second  order.  It  will  be  recalled  that 
the  hexagonal  prism  of  this  order  remains  unchanged,  geometrically, 
by  the  trigonal  hemihedrism,  page  59.  Hence,  as  in  the  preceding 
case,  each  face  belongs  alike  to  the  upper  and  lower  poles,  hemi- 
morphism  also  can  not  be  effective  in  the  production  of  new  forms. 
This  prism  is,  therefore,  still  apparently  holohedral. 

Its  symbols  are:     2a  :  2a  :  a  :  one,  ooP2,  \hh2ho\. 

Ditrigonal  prisms.  As  in  the  case  of  the  trigonal  prisms,  page 
59,  these  forms  remain  unchanged. 

(na  \  pa  :a\  ace]  ccPn 

The  symbols  are  :     -f^  I  -  -  -  I  ;  +     —  —  ,    \htko\\ 

QQ  Pn 


These,  together  with  the  trigonal  prisms,  are  not  to  be  dis- 
tinguished morphologically  from  those  of  the  ditrigonal  bipyramidal 
class.  They  are  apparently  hemihedral. 

Basal  pinacoids.  This  form  now  evidently  consists  of  two 
types,  upper  and  lower,  of  one  face  each. 

f  oca  :  oca  :  oca  :  c]  OP 

The  symbols  are:  -  -  •(«,/,•   :  -   u,    5oooi}; 

-  /,   1  0001  1  . 

2 

In  all  the  above  forms  the  crystallographic  axes  are  located  as 
in  the  ditrigonal  bipyramidal  class. 


DITRIGONAL    PYRAMIDAL    CLASS. 


73 


The  following  table  shows  the  principal  features  of  the  forms  of 
this  class: 


FORMS 

SYMBOLS 

Number  of  Faces 

Solid 
Angles 

1 
H 

Hexahedral 

Weiss 

Naumann 

Miller-Bravais 

Trigonal  Pyramids 
First  order 

_j_7y  _,_y  fa:OCa:a:mc] 

+  mPu 

4 
mP 

4 

\  holil  \ 
[hohl\ 
\ohhl\ 

"3 

J 
6 

I 

- 

-  '  —    (          4          \ 

Hexagonal  Pyramids 
Second  order 

j\2a  :  20,  :  a  :  mc\ 

mP2 

\hh2~hl\ 
\hh2h  l\ 

- 

I 
I 

2 

m  P2 

It,      [                 2                J 

2 

Ditrigonal  Pyramids 

\  U       \-l 

m  Pn 

4 

mPn 
~  u 
4 
mPn 

4 

\ihll\ 
\ihkl\ 

6 

1  '-(         4           \ 

Trigonal  Prisms 

(a  :  CO  a  :  a  :  OC<:1 

• 

r     OOP 

|  h  o  ~h  o  [ 
\  o  h  h  o  j 

Appar- 
ently of 
ditrigonal 
bipyra- 
midal 
class. 

2 

OCP 

-I 

2 

Ditrigonal  Prisms 

f«#  :  pa  \  a  :  mc^ 

,    &Pn 

(.   •  j  —  -    ) 

t                > 

2 

CCPn 

—  1                   2                 J 

2 

Hexagonal  Prism 
Second  order 

20.  :  2a  :  a  :  OO  c 

OO   P2 

\  h  h  2  ti  o  \ 

Apparently 
holohedral 

Basal  Pinacoids 

r   OP 

OP 

{  0  0  0  I   } 
\OOOI     \ 

I 

- 

- 

1                   2                     J 

Combinations.      The    mineral     tourmaline    furnishes    excellent 
combinations  of  the  above  forms. 


74 


HEXAGONAL   SYSTEM. 


In   the  accompanying   figures 

P  - 

214  and  215,  P  =  4-  —  u,  {  ion  }  ; 

4 

P  __  2P 

P  =   +     -  /,   Join  |  ;  o  =  --  w, 
4  4 

OP 

50221  i;   c  -       —I,    {oooij;    n  — 


t    =    + 


Fig.  215. 


u,  2131;  /  = 


OOP 


f . . 

lOIIOj 


H20 


4  2 

As  indicated  on  page  70,  these  forms  may  be  assumed  as 
derived  from  those  of  the  ditrigonal  scalenohedral  class,  i.  e.,  showing 
rhombohedral  hemihedrism,  and,  hence,  are  often  designated  as 
follows  : 


=          «;  c=  I 


ooR 
=  ooP2;   /  —  --  ;  o  =  —  2Ru. 


8.     HEXAGONAL  PYRAMIDAL  CLASS. 

(Pyramidal  Hemihedrism  with  Hemimorphism.) 

Symmetry.  This  class  possesses  but  one 
element  of  symmetry,  namely,  a  polar  axis  of 
hexagonal  symmetry,  as  shown  in  figure  216. 
This  is  obvious,  for  when  the  forms  of  the  hex- 
agonal bipyramidal  class,  page  65,  become 
hemimorphic,  the  principal  plane  and  center 
of  symmetry  are  of  necessity  lost.  Forms  of 
exactly  the  same  character  are,  moreover,  to 
be  derived  by  applying  hemimorphism  to  those 
of  the  hexagonal  trapezohedral  class.  Compare  figures  203  and  209. 

Forms.  The  bipyramids  of  the  first,  second  and  third  orders  of 
the  hexagonal  bipyramidal  class1}  now  consist  of  upper  and  lower 

1)  If,  however,  hemimorphism  be  applied  to  the  hexagonal  trapezohedral  forms,  the  hexagonal 
trapezohedrons  and  dihexagonal  prism  would  yield  forms  corresponding  precisely  to  the  pyramids  and 
prisms  of  the  third  order.  The  other  forms  being  apparently  holohedral  in  both  classes  would,  of 
course,  be  affected  alike  by  the  introduction  of  hemimorphism. 


Fig.  216. 


HEXAGONAL    PYRAMIDAL    CLASS. 


75 


pyramids.  The  prisms,  on  the  other  hand,  remain  unaffected  mor- 
phologically. The  basal  pinacoid,  of  course,  is  now  composed  of  an 
upper  and  lower  form. 

As  in  the  preceding  class,  these  are  now  open  forms.  Their 
position  in  respect  to  the  crystallographic  axes  is  the  same  as  in  the 
hexagonal  bipyramidal  class,  page  65. 

The  principal  features  are  given  in  the  following  table: 


FORMS 

SYMBOLS 

FACES 

Weiss 

Naumann 

Miller- 
Bravais 

Hexagonal 
Pyramids 
First  order 

.(a  :  OOa  :  a  :  mc\ 
u   / 

f        mP 
u 

mP 

{       ~l 

\ho~hl  \ 
\ho~hl\ 

1    ' 

I                    2                   J 

Hexagonal 
Pyramids 
Second  order 

,  (  20,  :  20,  :  a  :  me  } 

(       mP2  u 

{  hh2hl  \ 
\hh~2Jil\ 

6 

J 

m2p2 

I              2              J 

(               2         l 

Hexagonal 
Pyramids 
Third  order 

+  u   +  [{na:pa:a:mc} 

(        mPn 

\hi~kl] 
\hi~kl  \ 
\k1Ji  l\ 
\kihl  \ 

} 
6 

+     4     U 
+  mPnl 

mPn 
u 

4 

mPn  1 

(   ~     4 

1             4             J 

Hexagonal  Prism 
First  order 

a  :  OC  a  :  a  :  X  c 

00  P 

{ho~ho\ 

Appar- 
ently 
[  holohe- 
dral 

Hexagonal  Prisms 
Second  order 

2(i  :  20,  :  a  :  Qcr 

OOP  2 

\  hh~2Jil\ 

Hexagonal  Prisms 
Third  order 

fmz  :pa:a:VOc'\ 

f          OOPn 

00  Pn 

I              » 

{  hi~ko\ 
\kTko  \ 

1 
6 

±1                     2                   J 

Basal  Pinacoids 

,  fOC0  :  OCa  :  OC«  :  c] 

(      OP 
—  u 

(  ?< 

{  oooi  \ 

\  ooo7\ 

i  ' 

*'  *(                        2                        J 

Combinations.    By  means  of  etch  figures  crystals  of  a 
compounds    have  been  referred   to  this  class.      Among  the 


number  of 
minerals, 


76 


HEXAGONAL   SYSTEM. 


nepheline,  Na8Al8Si9O34,  is  to  be  mentioned.      Figure  217 
shows  a  crystal  of  strontium  antimonyltartrate,  Sr  (SbO)2 

p 


=  ~u,   {ion 


-/,    J202IL 


HEXAGONAL  TETARTOHEDRISMS. 

The  tetartohedral  forms  may  be  assumed  as  derived 
from  the  holohedral  by  applying  simultaneously  two  types 
Fig.  217.  of  hernihedrism,  page  55.     Thus,  by  combining  the  tri- 

gonal and  pyramidal  hemihedrisms,  the  trig-onal  tetarto- 
hedrism results,  figure  218.  A  second  type,  known  as  the  trapezo- 
hcdral  tetartohedrism,  is  the  result  of  the  combination  of  the 


Fig.  218. 


Fig.  219. 


Fig.  220. 


rhombohedral    and   trapezohedral  hemihedrism,   figure    219.       The 
simultaneous  application  of  the  rhombohedral  and  pyramidal  hemihe- 
drisms gives  rise  to  the  rhombohedral  tetartohedrism,  figure  220. 
These  three  are  the  only  types  possible,  for  other  combinations 
of  the  four  methods  of  hemihedrisms  do  not 
satisfy  the  conditions  of  tetartohedrism. 

9.     TRIGONAL    BIPYRAMIDAL    CLASS. 

(  Trigonal  Tetartohedrism. ) 

Symmetry,    This  class  possesses  two  ele- 
ments of  symmetry,  namely,  a  principal  plane 
Fig.  221  and  an  axis  of  trigonal  symmetry,  figure  221. 


TRIGONAL    BIPYRAMIDAL    CLASS. 


77 


Trigonal  b  i  p  y  r  a- 
mids  and  prisms  of  the 
first  order.  As  can  be 

readily  seen  from  figures 
222  and  223,  the  hexa- 
gonal bipyramids  and 
prism  of  the  first  order  now 
yield  trigonal  bipyramids 
and  prisms,  respectively. 
These  forms  do  not  differ 
morphologically  from 
those  of  the  ditrigonal  bipyramidal  class,  compare  figures  160  to  168. 
The  symbols  for  the  trigonal  bipyramids  of  the  first  order  are: 

a  :  oca  :  a  :  me]       .   mP 

—^~        -J,  +—  ,    \hohl\\   and 

a  :  ooa  :  a  :  me]          mP  —    , 

—          J,   -    —  ,    {ohhl\. 

Those  for  the  corresponding  trigonal  prisms  being: 

a  :  oca  :  a  :  ooc  ocP 

+  \hoho\\   and 


Fig.  222. 


Fig.  223. 


a  :  cca  :  a  : 


—  -,    \ohho\. 


Trigonal  bipyramids  and  prisms  of  the  second  order.    These 
forms  are  obtained  from  the  hexagonal  bipyramid  and  prism  of  the 


Fig.  224. 


Fig.  225. 


Fig.  226. 


Fig.  227. 


Fig.  228. 


Fig.  229. 


78 


HEXAGONAL    SYSTEM. 


second  order  as  can  be  seen  from  figures  224  to  229.  The  position 
of  these  forms  as  well  as  those  of  the  other  two  orders  in  respect  to 
the  crystallographic  axes  is  shown  in  figure  237. 

The  symbols  for  the  trigonal  bipyramids  of  the  second  order: 


( 2a  :  2a 

I 


me 


?;z  P  2 


— 
{h  h  2  hl\,  figure  226;    and 


:  2a  :  a  :  me 


,   -     — ^-,    J 2  hhhl\,  figure  224. 


Those  of  the  corresponding  trigonal  prisms: 

2a  :  2a  :  a  :  ooc]          ocP2 


,    \h1i2ho},  figure  229,  and 


\2a  :  2a  :  a  :  ooc]          Gc?2 

I  ,   -          — ,    | 2  h  h  h  o  \ ,   figure  227. 

(  2  J  2 

Trigonal  bipyramids  and  prisms  of  the  third  order.       The 

dihexagonal  bipyramid  now  furnishes  four  new  forms,  termed  trig- 
onal bipyramids  of  the  third  order,  while  the  dihexagonal  prism 
yields  the  corresponding  prisms.  Figures  230  to  235  show  the 
derivation  of  two  forms  of  the  bipyramid  and  prism.  Figures  236 
and  237,  however,  show  the  position  of  these  forms  in  respect  to  the 
crystallographic  axes  as  also  to  the  forms  of  the  other  orders. 


Fig.  230. 


Fig.  231. 


Fig.  232. 


Fig.  233. 


Fig.  234. 


Fig.  235. 


TRIGONAL   BIPYRAMIDAL   CLASS. 


79 


The  symbols  of  the  bipyramids  are: 

Positive  right, 

{ na  :  pa  :  a  :  me 
+  r[~       ~2~ 

Positive  left, 

;  ^na-.pa-.a-.mc^  .    + 


.   {hikl},   figure  232. 


.    —  —   , 

\kihl\,   figure  230. 


Negative  right, 


I  2 

Negative  left, 

{ na  :  pa  :  a  :  me 

Those  of  the  prisms  are: 
[  na  :  pa  :  a  :  G£c 
2 


_ 

J  4 


-I-     -;    \khil\. 


;    \hiko\i  figure  235. 
;    \kiho\,    figure  233. 


na:pa:a:aoc]  QcPw     (.,-r    i 

r\-      -j-        I;  —  r~^— ;  jiA^o}. 


Fig.  236. 


Fig.  237. 


The  basal  pinacoid,  as  is  obvious  from  figures  223,  ^28  and 
remains  unaltered. 


80 


HEXAGONAL    SYSTEM. 


The  following  table  summarizes  the  principal  features  of  each 


form. 


No  representative  of  this  class  has  yet  been  discovered. 


FORMS 

SYMBOLS 

Number  of  Faces 

Solid    . 
Angles 

Trihedral 

H 

Weiss 

Naumann 

Miller-Bravais 

Trigonal  Bipyramids 
First  order 

. 

f#  :  QC  #  :  a  :  mc^ 

f+¥ 

1  _^«/) 

L      ^ 

{  A^  J/} 
J0AA/  | 

1- 

2 

3 

±(      *       I 

Trigonal  Bipyramids 
Second  order 

_!_  f.?a  :  2a  \  a  :  ntc\ 

mP2 

{  AA  J^/  \ 

\  2hhhl  \ 

Q 

2 

3 

mPz 

I           J~ 

—  I            2           J 

Trigonal  Bipyramids 
Third  order 

_j_^   +  ,\na\pa\a\mc'\ 

'+r=S 
4 

mPn 

\hiJl\ 
\kthl  \ 
\  ih~kl\ 

\  kJTi  \ 

] 
Q 

J 

2 

3 

\+    *  - 

mPn 

-'-  [          4           J 

4 
tmPn 

;            ^ 

Trigonal  Prisms 
First  order 

^{a:  OOa  :a:OOc] 

L^P 

00  P 

l~"T^ 

\hoho\ 
\oh~ho] 

3 

-L             2             J 

Trigonal  Prisms 
Second  order 

_,_  \2a\2a  :  a:  OOc] 

f     oc/7^ 

\II~2   0\ 

{  ^77o\ 

3 

.  +     ^ 

00  P2 
2 

—  I                  2                    J 

Trigonal  Prisms 
Third  order 

,  r    |  i\na\pa\a\&c\ 

f       oc  />,/ 

\hi~ko  \ 
\  k~i~ho   \ 
\  ihlio\ 
\  kh^o  \ 

3 

J 

- 

- 

4 
,<XPn 

'              4 
OOPn 

~  1             4          J 

4 
,00  Pn 

I                 4f 

Basal  Pinacoid 

OCa  :  OCrt  :  OCtf  :  <: 

6>r 

\0  0  0  I   } 

2 

- 

- 

TRIGONAL    TRAPEZOHEDRAL    CLASS. 


81 


10.     TRIGONAL  TRAPEZOHEDRAL  CLASS.*) 

(  Trapezohedral  Tetartohedrism. ) 

Symmetry.  A  principal  axis  of  trigonal 
symmetry  and  three  polar  axes  of  binary 
symmetry  are  the  only  elements  in  this 
class,  figure  238. 

Rhombohedrons    of   the    first  order. 

From  the  hexagonal  bipyramid  of  the  first 
order,  two  rhombohedrons  of  the  same 
order  result  as  can  be  seen  from  figure  239. 
These  forms  are  identical,  morphologically, 
with  those  of  the  ditrigonal  scalenohedral 

class,  page  60. 

,    {a:Cioa:a: me 
Their  symbols  are:  Hr    '        

+  mR,  \ho7il\\  —  mR,  \oh7il\.2} 

Trigonal  bipyramids  of  the  second 
order.  The  hexagonal  bipyramids  of  the 
second  order,  figure  240,  yield  two  trigonal 
bipyramids  of  this  order.  They  are  identi- 
cal, geometrically,  with  those  of  the  trigonal 
bipyramidal  class,  page  77. 3 

I \ 2a\  ia  \a\mc 
The  symbols  are: 


Fig.  238. 


Fig.  239. 


Trigonal  Trapezohedrons.  The  dihex- 
agonal  bipyramids  yield  four  new  forms, 
each  bounded  by  six  faces,  which  in  the 
ideal  development  are  equal  trapeziums. 
Figure  242  shows  the  application  of  the  Fig.  240. 

trapezohedral  tetartohedrism  so  as  to  give 

rise  to  the  positive  right  trigonal  trapezohedron,  figure  243.     Fig- 
ure 241  shows  the  positive  left  form.      These  are  enantiomorphous. 

1)  The  trapezohedral  group  of  Dana . 

2)  Sometimes  the  two  Greek  letters  K  T  are  placed  before  the  Miller  indices.     See  footnote  on 
page  40. 

3)  Sometimes  designated  as  right  and  left. 


82 


HEXAGONAL    SYSTEM. 


The  trigonal  trapezohedrons  may  also  be  conceived  as  derived 
from  the  scalenohedron  by  the  subsequent  application  of  hemihe- 
drism.  This  is  shown  by  figure  245,  a  negative  scalenohedron,  which 
now  yields  the  negative  right  (figure  246)  and  negative  left  (figure 
244)  trigonal  trapezohedrons,  respectively. 


Fig.  244.  Fig.  245.  Fig.  246. 

The  symbols  may  be  written  as  follows: 
i)     Positive  right, 

na\pa\a\mc\      ,    ^mPn  .  r  ¥l 

-j-  j  ,  -h  r -— ,    \hikl\>    figure  243- 


TRIGONAL    TRAPEZOHEDRAL    CLASS. 


83 


«)     Positive  left, 

f  na  :  pa  :  a  :  me]       .     .  mPn 

-h  / ,   H-  /  - 

L  4  J  4 

3)  Negative  right, 

na  :  pa  :  a  :  me]  mPn 

± *• 

4  4 

4)  Negative  left, 

na  :  pa  :  a  :  me]  _     mPn 
4              J'  4     ' 


—  r 


\k  i  h  l\ ,   figure  241. 


i  h  k  /},    figure  246. 


k  h  i  /},    figure  244. 


Forms  i  and  2,  3  and  4  are  among  themselves  enantimorphous, 
while  i  and  3,  2  and  4  are  congruent. 

The  polar  axes  of  binary  symmetry  bisect  the  zigzag  edges. 

Trigonal  prisms  of  the  second  order. 

Figure  247  shows  that  the  hexagonal  prism 
of  this  order  yields  two  trigonal  prisms. 
These  are  similar  to  those  of  the  trigonal 
bipyramidal  class,  page  77. 


+ 


The  symbols  are : 
-,  \hh2ho\\  — 


OCP2 


2  h  h  h  o 


Ditrigonal  Prisms.     The  dihexagonal  Fig  247. 

prism  now  gives  rise  to  two  ditrigonal  prisms 

as  shown  in  figures  248,  249  and  250.  These  are  similar,  morpho- 
logically, to  those  of  the  ditrigonal  bipyramidal  class,  page  59,  but 
differ  from  them  in  respect  to  the  position  of  the  crystallographic  axes. 
A  comparison  of  figures  251  with  172  and  174  shows  the  difference. 


Fig.  248. 


Fig.  249. 


Fig.  250. 


84 


HEXAGONAL    SYSTEM. 


,~,                                     ,    \na  :  -ha  :  a  :Ooc 
The   symbols   are :      Hr  - 1  ;  -f- 

figure  250;  -  -,  \ki7io\,  figure 


,    \hiko\t 


Fig.  251. 


Fig.  252. 


The  other  forms, 
the  hexagonal  prism 
of  the  first  order  and 
the  basal  pinacoid, 
remain  unchanged, 
figure  252. 

The  important 
features  of  this  class 
are  given  in  the  fol- 
lowing table: 


FORMS 

SYMBOLS 

Number  of 
Faces 

Solid 
Angles 

|1 

si 

H^ 

Weiss 

Naumann 

Miller-Bravais 

Rhombohedrons 
First  order 

fa:QOa:a:  me 

f    +mR 
\   —  mR 

{A  o  A  /} 

{oh  hi) 

\ 

App 
ty  \ 
hed 

2 

irent- 
emi- 

3 

—    L                    2 

Trigonal  Bipyramids 
Second  order 

_L_  \2a  :  20,  :  a  :  me] 

f        mP2 

{2h'hhl} 

\      ^  ^ 

—    I                   2                   } 

Trigonal  Trapezo- 
hedrons 

fna:pa:a:mc] 

f           mPn 

{hill} 
{kihl} 
[i  hkl} 

6 

J 

2 
6 

4 

mPn 

±'•±•1     4     J 

4 
j  mPn 

4 

Hexagonal  Prism 
First  order 

a  :  CQa  :  a  :  CCc 

OOP 

\hoHo\ 

6 

Appj 
he 

rent- 
holo- 

iral 

Trigonal  Prisms 
Second  order 

OO  P2 

{hh'zho} 
\2hh~ho\ 

F 

2 
2 

—  (.                  2                   | 

Ditrigonal  Prisms 

_l_   i         L£___! 

\         CCPn 

\hiko\ 
\kTJto] 

6 

2 
j          OOPn 

{           2 

-(      1       J 

Basal  PJnacoid 

CGa  :  CCa  :  OOa  :  c 

OP 

\O  O  O  I\ 

2 

Apparent- 
ly holo- 
hedral 

*)    Often  designated  as  right  and  left  forms. 


UNIVERSITY    1 

C4L!? 


TRIGONAL    RHOMBOHEDRAL    CLASS. 


85 


Combinations.     Quartz,    SiO2 ,    and    Cinnabar,    HgS,     furnish 
excellent  examples  of  minerals  crystallizing  in  this  class. 


Fig.  253. 


Fig.  254. 


Fig.  255. 


Figures   253   and   254.        a  —  oo  R  j  ioio(  ;    r  =  +  R  {  101  1  j  ; 

2P2 


=  —R{oiu};  $  (figure  254)  =    +—  -{Il2i},   (figure253) 


21  fi};   *  (figure  254)     =  +  r 


Figure  255.     r  = 


-     [  JS^il,    (figure  253)    +   /- 
4  4 

;  w  =  OoR{ioio};  r  =  -f  R{  ion  |  ; 


-IP4 

g-  =—  iR|oii2(;   n'  =  —  2R|o22ii;  and  x  =  +  /  ^-^  18355.  J  Cin- 
nabar. 

//.     TRIGONAL  RHOMBOHEDRAL   CLASS.1) 

( Rhombohedral   Tetartohedrism. ) 

Symmetry.  This 
class  possesses  a  prin- 
cipal axis  of  trigonal 
symmetry.  A  center  , 
of  symmetry  is  also 
present,  figure  256. 

Rhombohed  r  o  n  s 

of  the  first  order.  The  Fig.  256. 

hexagonal    bipyramid 

of  the    first  order,   figure  257,   now  yields  two 
rhombohedrons  which  are  identical,  geometric- 


Fig.  257. 


The  tri  rhombohedral group oi  Dana. 


HEXAGONAL    SYSTEM. 


ally,    with  those   of    the  ditrigonal   scalenohedral    class,    page   60, 
figures  179  and  181. 
The  symbols  are: 


I 


f  a  :  00  a  :  a  :  mc\ 


\  —  mR{o/i/il\. 


Rhombohedrons  of  the  second  order.  Figure  259  shows  the 
application  of  rhombohedral  tetartohedrism  on  the  hexagonal  bipyr- 
amid  of  the  second  order.  This  form  now  yields  the  rhombohedrons 
of  the  second  order. 


Fig.  258. 

The  symbols  are: 
,    \2a  :  2a  :  a  :  me 


Fig.  259. 


Fig.  260. 


+  — - —  \hh2hl\   figure  260; 


1 2  h  h  h  l\   figure  258. 

Rhombohedrons  of  the  third  order.      From   the   dihexagonal 
bipyramid  four  rhombohedrons  of  the  third  order  result.     Two  of 


Fig.  261. 


Fig.  262. 


Fig.  263. 


TRIGONAL    RHOMBOHEDRAL    CLASS. 


87 


these  are  shown  in  figures  261,  262  and  263.  The  position  of  these 
rhombohedrons  with  respect  to  the  crystallographic  axes  is  shown  by 
figures  236  and  237,  which  also  indicate  the  relation  existing  between 
the  rhombohedrons  of  the  three  orders. 


The  symbols  are: 

Positive  right, 

na  :  pa  :  a  :  me} 

4 
Positive  left, 

f  na  :  pa  :  a  :  me 

'  4 
Negative  right, 

f  na  :  pa  :  a  :  me 

~~4~~ 
Negative  left, 

C  w«  :  ^>«  :  «  :  w<7 

\k~hil\. 


mPn     ( ,  .  r ,)     ^ 

r-    — ,    \hikl\,    figure  263. 
4 

,wPw 


—  r 


4 
mPn 


—  r 


\ki hl\,    figure  261. 
\ihkl\. 


mPn 

4    > 


Hexagonal  prisms  of  the  third  order. 

Figure  264  shows  that  the  dihexagonal  prism 
now  yields  two  hexagonal  prisms  of  the  third 
order.      Compare  figures  205  and  207. 
The  symbols  are: 


± 
oo  P^ 


\na  \pa  :a: 


ccPn 


r 

\hiko\-, 


Fig.  264. 


\kiho\. 


Other     forms. 

The  remaining  holo- 
hedrons  are  unchanged 
by  this  tetartohedrism, 
see  figures  265  and 
266. 


Fig.  265. 


Fig.  266. 


88  HEXAGONAL   SYSTEM. 

The  principal  features  may  be  tabulated  as  follows: 


FORMS 

SYMBOLS 

V) 
V 

u 
cS 

b 

Solid  Angles 

Weiss 

Naumann 

Miller- 
Bravais 

Trihedral 

Rhombo- 
hedrons 
First  order 

faiocaia:  mc^\ 

[   +;»/? 
-mR 

|A0£/} 

6 

Apparently 
of  ditrigonal 
scalenohe- 
dral  class 

±1       *       J 

|o^A/} 

Rhombo- 
hedrons 
Second  order 

\2a  :  2a  :  a  :  me] 

' 

,  wft 

{hh~2lil\ 
\2hhh  l\ 

1 
-6 

J__ 
6 

21>-h  6 

1 
mP2 

—  I                    2                   J 

2 

Rhombo- 
hedrons 
Third  order 

,  r    ,  rfna-.pa-.a-.tnc} 

mPn 

\hikl\ 
\k~ih  l\ 
\ihkl\ 
{khTl\ 

2l>  +  6 

1      / 

4- 
mPn 

I    / 
^ 

mPn 

~L    (            4      •      J 

7 

4 
^/^^ 

/ 
4 

Hexagonal 
Prism 
First  order 

a  :  oca  :  a  :   aoc 

OOP 

\IOIO\ 

[6 
J 

Apparently 
holohedral 

Hexagonal 
Prism 
Second  order 

2a  :  2a  :  a  :  occ 

OOP2 

\II20} 

Hexagonal 
Prisms 
Third  order 

,    \  na  :  pa  :  a  :  oo  c  "] 

.   ocPn 

.  +  ~ 

ocPn 

\hiko\ 
\ki7io} 

I6 

JL 

2 

Apparently 
of  hexagonal 
bipyramidal 
class 

1l        *       J 

I 

Basal 
Pinacoid 

ooa  :  oca  :  oca  :  c 

OP 

\000l] 

Apparently 
holohedral 

Three  equal  edges. 


TRIGONAL    PYRAMIDAL    CLASS. 


89 


Combinations.    Dolomite,  CaMg(CO3)2,  and  dioptase,  J^CuSiO^ 
crystallize  in  this  class. 

Figure  267.   m  =  OcP2,  j  1 120} ; 
r  =  —  2R,  5 022 1  j ;  5  =  —  /  *    — , 


{18. 17. 1.8},  dioptase. 

Figure  268.     m  =  4R,  {4041 } ; 

•VP2 

r  =   R, .   jioiij;   ?z   =   •       > 

5i6.8-8.3j1>;  *:  =  OR,  {oooi},    dol- 
omite. 


Fig.  268. 


12.     TRIGONAL  PYRAMIDAL  CLASS. 

[Ogdohedrism.  "] 

Trigonal  Tetartohedrism  with  Hemimorphism.) 

Symmetry.      The  only  element  of  symmetry  is  a  polar  axis  of 
trigonal  symmetry  parallel  to  the  c  axis,  figure  269. 

^Mp 


Forms.  When  hemimorphism  becomes  effective  on  forms  of 
the  trigonal  bipyramidal  class,  page  76,  it  is  evident  the  various 
bipyramids  and  the  basal  pinacoid  give  rise  to  upper  and  lower  forms. 
The  prisms,  however,  remain  unchanged  since  they  belong  alike  to 
the  upper  and  lower  poles. 

The  principal  features  of  these  forms  are  given  in  the  following 
table: 


Rhombohedron  of  the  second  order. 


90 


HEXAGONAL   SYSTEM. 


FORMS 

SYMBOLS 

FACES 

Weiss 

Naumann 

Miller-Bravais 

Trigonal  Pyramids 
First  order 

[-  a  :  OCa  :  a  :  me  ] 

J    ^'u 

4 

-\   ~l 

mP 
—  u 

[-?' 

\hoHl\ 

\ho~hl\ 

3 

.- 

1  I              4              J  "' 

\ok~hl] 

\oh~hl  \ 

Trigonal  Pyramids 
Second  order 

_,_  f  20  :  20.  :  a  :  me  ~] 

f+ST" 

+  fT' 
_«^a 

4 
mPa 

l~^~  ' 

\htT2~hl  \ 
\hh~2~hl  \ 

\2hhhl} 

\  2  hhhl  \ 

3 

J 

~  I              4              J  "' 

Trigonal  Pyramids 
Third  order 

+r  _._/  r  ««:>«:  a  :«^        , 

r      w^w 

+  r-y-  « 

w/^n 
+  'T-« 

mPn 

-r-T« 

,  mPn 

-i  _„ 

+-T' 

+  /WL^/ 

6* 
-  r  wPn  / 

"y~^ 

-/  "L^?/ 

<y 

\hfll\ 
\kihl\ 
\ihkl\ 
\khil\ 
\ihkJ\ 
1*177} 
\ki17\ 
\kik7] 

1 
-            3 

J 

±r'±f  [                «?                 J    «•' 

Trigonal  Prisms 
First  order 

_^   f  a  :  QCa  :  a  :  QCO 

r+«? 

QCP 

1           * 

|  fOfO^ 
|  OI10  | 

Apparently 
of  trigonal 
j    bipyramidal 
class 

±l        '        J 

Trigonal  f'risms 
Second  order 

j.  f  2a  :  20  :  a  :  QCc  ^ 

r     QDA 

\rno\ 
{mo} 

J        '            2 
<X>P2 

"I               *               J 

( 

Trigonal  Prisms 
Third  order 

f««:M:a:   QCc  'j 

f+'^ 

QC/>n 

\hi~ko  \ 
\kiho  \ 
\hiko  \ 
\kh~io  \ 

^ 
r    ODP« 

'^l                 ^                J 

4 

_/  OP** 

1              * 

Basal  Pinacoids 

f  QCfl  :  QCa  :  QCa  :  c  1 

f(?/> 

•  •  M 

j^f 

\OOOI\ 

\ooo7} 

1 

1                   *                   J"' 

TRIGONAL   PYRAMIDAL   CLASS. 

P 


Combination.     Figure  270.     r=  + 


=  -        -  w,     J022I};     ^=—/, 


_ 
2134}.       Sodium 


joooij;     /=  — 
periodate,  NaIO4  +  3H2O. 


91 


Fig.  270. 


TETRAGONAL  SYSTEMS 


Crystallographic  Axes.  The  tetragonal  system  includes  all 
forms  which  can  be  referred  to  three  perpendicular  axes,  two  of  which 
are  equal  and  lie  in  a  horizontal  plane. 
These  are  termed  the  secondary  or  lateral 
axes  and  are  designated  as  the  a  axes.  Per- 
pendicular to  the  plane  of  the  lateral  axes  is 
the  -principals?  c  axis,  which  may  be  longer 
or  shorter  than  the  a  axes.  The  axes, 
which  bisect  the  angles  between  the  a  axes, 
are  the  intermediate  axes.  They  are 
designated  as  the  b  axes  in  figure  271. 

Crystals  of  this  system  are  held  so  that 
the  c  axis  is  vertical  and  one  of  the  a  axes 
is  directed  towards  the  observer. 

Since  the  lengths  of    the  a  and  c  axes 

differ,  it  is  necessary  to  know  the  ratio  existing  between  these  axes, 
that  is,  the  axial  ratio,  as  was  the  case  in  the  hexagonal  system. 
Compare  pages  7  and  43. 


-c 

Fig.  271. 


Classes  of  Symmetry. 

symmetry,  as  follows: 


This  system  embraces  seven  classes  of 


1 .  Ditetragonal  bipyramidal  class 

2.  Ditetragonal  pyramidal  class 

3.  Tetragonal  scalenohedral  class  1 

4.  Tetragonal  bipyramidal  class      \ 

5.  Tetragonal  trapezohedral  class  j 

6.  Tetragonal  pyramidal  class 

7.  Tetragonal  bisphenoidal  class 


(Holohedrism.) 

{ Holohedrism  and} 
\hemimorphism.      J 

(Hemihedrism) . 

[Hemihedrism  and\ 

\hemimorphism.      J 

( Tetartohedrism). 


Also  termed  quadratic  or  pyramidal  systei 


[92] 


DITETRAGONAL    BIPYRAMIDAL    CLASS. 


93 


No  representative 


Classes  i ,  3  and  4  are  the  most  important, 
of  class  7  has  yet  been  observed. 

DITETRAQONAL  BIPYRAMIDAL   CLASS.1) 

( Holohedrism. ) 

Symmetry,  a)  Planes*  In  this  class 
there  are  five  planes  of  symmetry.  The 
plane  parallel  to  the  plane  of  the  secondary 
and  intermediate  axes  is  termed  the 
principal  plane.  The  vertical  planes  includ- 
ing the  c  and  one  of  the  a  axes  are  called 
the  secondary  planes,  while  those  which 
include  one  of  the  b  axes  are  termed  the 
intermediate  planes,  figure  272. 

The  principal  and  secondary  planes 
divide  space  into  eight  equal  parts,  termed 
octants.  The  five  planes,  figure  272, 
divide  it  into  sixteen  equal  sections.  The 
five  planes  may  be  designated  as  follows: 

i  Principal  -h  2  Secondary  +  2  Inter- 
mediate =  5  Planes. 

b)  Axes.     Parallel  to  the  c  axis,  there 
is  an  axis  of  tetrag-onal  symmetry.     The 
axes  parallel  to    the    secondary  and    inter- 
mediate   axes     possess    binary     symmetry, 
figure  273.      These  may  be  written:       i| 
-}-2«+20:=5  axes. 

c)  Center.     A  center  of  symmetry  is 
also  present  in  this  class.     These  elements 
of  symmetry  are  shown  in  figure  274,  which 
represents  the  projection  of  the  most  complex 

form  upon  the  principal  plane  of  symmetry.  Flg*  2'4* 

Tetragonal  bipyramid  of  the  first  order.      This   form   is   anal- 
ogous to  the  octahedron  of  the  cubic  system,  page  19.       But  since 


Fig.  273. 


1)    The  normal  group  of  Dana. 


94 


TETRAGONAL    SYSTEM. 


Fig  275. 


the  c  axis  differs  from  the  secondary  axes,  the 
ratio  must  be  written  (a  :  a  :  c\  which  would 
indicate  the  cutting  of  all  three  axes  at  unit 
distances,1)  figure  275.  But  since  the  inter- 
cept along  the  c  axis  may  be  longer  or  shorter 
than  the  unit  length,  the  general  ratio  would 
read  (a  :  a  :  me),  where  m  is  some  value 
between  zero  and  infinity.  Like  the  octahe- 
dron, this  form,  the  tetragonal  bipyramid,^ 
is  bounded  by  eight  faces  which  enclose  space. 
The  faces  are  equal  isosceles  triangles  when 
the  development  is  ideal. 


The  Naumann  and  Miller  symbols  are: 

P,  |  in};   or  raP,  \hhl\. 


The  principal  crystallographic  axis  passes  through  the  two  tetra- 
hedral  angles  of  the  same  size,  the  secondary  axes  through  the  other 
four  equal  tetrahedral  angles,  while  the  intermediate  axes  bisect  the 
horizontal  edges. 

Tetragonal  Mpyramid  of  the  second 
order.  The  faces  of  this  form  cut  the  c 
and  one  of  the  a  axes,  but  extend  parallel 
to  the  other.  The  parametral  ra,tio  is, 
therefore,  a  :  O0«  :  me.  This  requires 
eight  faces  to  enclose  space  and  the  form 
is  the  bipyramid  of  the  second  order, 
figure  276.  The  symbols  according  to 


Fig.  276. 


Naumann  and  Miller  are:    raPoc,   \ohl\. 

In  form  this  bipyramid  does  not 
differ  from  the  preceding,  but  they  can 
be  readily  distinguished  on  account  of 

their  position  in  respect  to  the  secondary  axes.  In  this  form,  the 
secondary  axes  bisect  the  horizontal  edges  and  the  intermediate 
axes  pass  through  the  four  equal  tetrahedral  angles.  This  is  the 


1)  Indicating  a  unit  form,  compare  page  7. 

2)  The  more  the  ratio  a  :  c  approaches  1 : 1,  the  more  does  this  form  simulate  the  octahedron. 
This  tendency  of  forms  to  simulate  those  of  a  higher  grade  of  symmetry  is  spoken  of  as  pseudo symmetry. 


DITETRAGONAL    BIPYRAMIDAL    CLASS. 


95 


Fig.  277. 


opposite  of  what  was  noted  with  the  bipyramid  of  the  first  order, 
compare  figures  275  and  276.  Hence,  the  bipyramid  of  the  first 
order  is  always  held  so  that  an  edge  is  directed  toward  the  observer, 
whereas  the  bipyramid  of  the  second  order  presents  a  face.  In  both 
bipyramids  the  principal  axis  passes  through  the  two  equal  tetra- 
hedral  angles. 

Di tetragonal  bipyramid.  The  faces  of 
this  bipyramid  cut  the  two  secondary  axes  at 
different  distances,  whereas  the  intercept  along 
the  c  axis  may  be  unity  or  me.  Sixteen  such 
faces  are  possible  and,  hence,  the  term  ditet- 
rag-onal  bipyramid  is  used,  figure  277. 1} 

The  symbols  are: 

(a\na:mc\    mPn,    \hkl\. 

Since  the  polar  edges2)  are  alternately 
dissimilar  it  follows  that  the  faces  are  equal, 
similar  scalene  triangles.  The  ditetragonal 
bipyramid  possessing  equal  polar  edges  is  crystallographically  an 
impossible  form,  for  then  the  ratio  a  :  na  :  me  would  necessitate  a 
value  for  n  equal  to  the  tangent  of  67°  30',  namely,  the  irrational 
value  2.4i42  +  .3) 

From  the  above  it  follows  that  when  n  is  less  than  2.41424-  the 
ditetragonal  bipyramid  simulates  the  tetragonal  bipyramid  of  the  first 
order,  and  finally  when  it  equals  I,  it  passes 
over  into  that  form.  On  the  other  hand,  if  n 
is  greater  than  2.4142+  it  approaches  more 
the  bipyramid  of  the  second  order,  and  when 
it  is  equal  to  infinity  passes  over  into  that 
form.  Hence,  n  >  I  <  oo.  Figure  278  illus- 
trates this  clearly.  It  is  also  to  be  noted, 
that  when  n  is  less  than  2.4142-!-  the  second- 
ary axes  pass  through  the  more  acute  angles, 
whereas,  when  n  is  greater  than  2.4142  + 
they  join  the  more  obtuse.  In  figure  278,  outline  I  represents  the 
cross-section  of  the  tetragonal  bipyramid  of  the  first  order,  2  that 


Fig.  278. 


*)  Compare  figure  24,  page  10. 
3)  Compare  footnote,  page  48. 
3)  See  also  page  40. 


96 


TETRAGONAL    SYSTEM. 


of  the  second  order,  and  3,  4  and  5  ditetragonal  bipyramids  where  n 
equals  f ,  3,  and  6,  respectively. 

Tetragonal  prism  of  the  first  order. 

If  the  value  of  the  intercept  along  the  c 
axis  in  the  tetragonal  bipyramid  of  this 
order  becomes  infinity,  the  number  of  the 
faces  of  the  bipyramid  is  reduced  to  four 
giving  rise  to  the  tetragonal  prism  of 
the  first  order,  figure  279.  This  is  an 
open  form  and  possesses  the  following 
symbols: 

(a  :  a  :  aoc),    ooP,  |  noj. 

The  secondary  axes  join  opposite 
edges,  hence,  an  edge  is  directed  toward 
the  observer. 


1_ 


1 
JL-. 


Fig.  280. 


Fig.  281. 


Tetragonal  prism  of  the  second 
order.  The  same  relationship  exists 
between  this  form  and  its  corresponding 
bipyramid  as  was  noted  on  the  preceding 
form. 

The  symbols  are: 
(a:  oca:  ooc),    ooPoo,  jioo}. 

This  is  also  an  open  form  consist- 
ing of  four  faces,  figure  280.  The  sec- 
ondary axes  join  the  centers  of  oppo- 
site faces.  Hence,  a  face  is  directed 
toward  the  observer. 

Ditetragonal  prism.  As  is  obvious, 
this  form  consists  of  eight  faces  possess- 
ing the  following  symbols: 

a:na:Ooc,     OoPw,    \hko\. 

What  was  said  on  page  94  concern- 
ing the  polar  angles  and  the  position  of 
the  secondary  axes  applies  here  also. 
Figure  281  represents  a  ditetragonal 
prism. 


DITETRAGONAL   BIPYRAMIDAL   CLASS. 


97 


Basal  Pinacoid.  This  form  is  similar  to  that  of  the  hexagonal 
system,  page  51.  It  is  parallel  to  the  secondary  axes  but  cuts  the 
c  axis.  The  symbols  may  be  written  : 

(ooa  :  oca  :  c],    OP,    jooif. 

This  form  consists  of  but  two  faces.  They  are  shown  in  combi- 
nation with  the  three  prisms  in  figures  279,  280,  and  281. 

These  are  the  seven  simple  forms  possible  in  the  tetragonal 
system.  The  following  table  shows  the  chief  characteristics  of  each. 


FORMS 

SYMBOLS 

I 
I 

Solid  Angles 

Tetrahedral 

Octahedral 

Weiss 

Nautnann 

Miller 

Unit  Bipyramid 
First  order 

a  :  a  :  c 

P 

{///} 

8 

2  +  4 



Modified  Bipyramids 
First  order 

a  :  a  :  me 

mP 

\hhl\ 

8 

2  +  4 



Bipyramids 
Second  order 

a  :  oca  :  me 

mPoc 

\hol\ 

8 

2  +  4 



Ditetragonal 
Bipyramids 

a  :  na  :  me 

mPn 

\hkl\ 

16 

4  +  4 

2 

Prism 
First  order 

a  :  a  :  occ 

ocP 

\IIO\ 

4 

— 

—  ' 

Prism 
Second  order 

a  :  oca  :  occ 

ocPoc 

\IOO\ 

4 

— 

.  —  . 

Ditetragonal  Prisms 

a  :  na  :  occ 

ocPn 

\hko\ 

8 

— 

— 

Basal  Pinacoid 

oca  :  oca  :  c 

OP 

\OOI\ 

2 

— 



TETRAGONAL   SYSTEM. 


Relationship  of  forms.     This  is  clearly  expressed  by  the  follow- 
ing diagram.     Compare  pages  23  and  52. 


oca  :  oca  :  c 


a  :  oca  :  me 


a  :  oca  :  occ 


a  :  7ia  :  me 


a  :  na  :  Oc  c 


a  :a  :  occ 


Combinations.      Some  of  the  more  common  combinations  are 
illustrated  by  the  following  figures: 


Fig.  282. 


Fig.  284. 


Fig.  285. 


Figures  282  to  286.  m  ^ooPjuoj,  p  —  Pjin},  a  = 
OoPocJioo},  u  =  3PS33M.  v=~-  2P|22i|,  and  x  -•  3P3J3ii}. 
These  combinations  are  to  be  observed  on  zircon,  ZrSiO4. 


Fig.  286 


Fig.  288. 


Fig.  289, 


DITETRAGONAL    PYRAMIDAL    CLASS. 


Figure    287.       m<  =    OoP 
JioiJ,  and  5  =  Pjiiij. 


uoJ,     a   =    OoPoo 
Cassiterite,  Sn2O4. 


IOO 


99 


e  = 


Figures    288    and    289.      p  =  P\m 


2  = 

|P|ii3},    i;=|PJii7},     w=OoP{uo},     a  =  ocPoojioo},    and 
e  =  P  oo  | 101  \.      These  combinations  occur  on  Anatase,  TiO2. 


,     c  =  OPjooij, 
a  =  ooP  oo  j  looi 


2.     DITBTRAQONAL  PYRAMIDAL   CLASS.1) 


(Holohedristn  with  Hemimorphism.} 

Symmetry.  Since  hemimorphism  is  effective  on  forms  of  this 
class,  the  principal  plane  and  the  four  axes  of  binary  symmetry  dis- 
appear. The  remaining  elements  are,  therefore,  two  secondary  and 
two  intermediate  planes  and  a  polar  axis  of  tetragonal  symmetry, 
figure  290. 


Fig.  290. 


Forms.  The  forms  of  this  class  are  analogous  to  those  of  the 
dihexagbnal  pyramidal  class,  page  53,  and  the  ditetragonal  bipyr- 
amids  and  tetragonal  bipyramids  of  the  first  and  second  orders  as  well 
as  the  basal  pinacoid  are  now  to  be  considered  as  divided  into  upper 
and  lower  forms.  For  example,  the  ditetragonal  bipyramid  now 
yields  an  upper  and  a  lower  ditetragonal  pyramid,  each  consisting 
of  eight  faces.  The  three  prisms,  however,  remain  unaltered;  they 
are  apparently  holohedral. 


*)     The  hemimorphic  group  of  Dana. 


100  TETRAGONAL   SYSTEM. 

The  principal  features  are  summarized  in  the  following  table: 


FORMS 

SYMBOLS 

in 

V 

% 

Solid  Angles 

Tetrahedral 

Octahedral 

Weiss 

Naumann 

Miller 

Upper 
and  Lower 
Pyramids 
First  order 

\a\a\  me]  u 

f   mP 
u 

2 

mP  1 

I 

I///J 

\i,7\ 

\hol\ 
\hol\ 

1 

•   4 

I 

[           2           \l 

2 

Upper 
and  Lower 
Pyramids 
Second  order 

\  a  :  ooa  :  mc\  u 

mPoo 

•   4 

I 

u 

2 

mPao  l 

[      *      \i 

I 

2 

Upper 
and  Lower 
Ditetragonal 
Pyramids 

(a  :  na  :  me]  u 

m  Pn 

\   * 

mPn 

\iiki\ 
\hkl\ 

8 



I 

1      *      \i 

I      * 

Tetragonal 
Prism 
First  order 

a  :  a  :  cc'c 

OOP 

\uo\ 

4 

4 

8 

j 

Appar- 
ently 
holohedral 

Tetragonal 
Prism 
Second  order 

a  :  oca,:  006 

oo  Poo 

\IOO\ 

Ditetragonal 
Prisms 

a  :  na  :  GCC 

ccPn 

\hko\ 

Upper 
and  Lower 
Basal  Pinacoids 

roo#  :  oca  :  c]  u 

r  OP 

\IOO\ 

\OOI\ 

•    i 



u 

1  °pl 

{               *               \l 

2     l 

DITETRAGONAL    PYRAMIDAL    CLASS. 


101 


r 
ip 

Combinations.      Figure  291.     x  =  ^—  u< 


|n3; 

Silver  fluoride,  AgF.H2O. 

Figure  292.     a  =   OoP  00 


w  =    -   /, 


100 


Fig.  291. 


Ill 


OP 


^r  <  )  X         7 

c   =  —    u>   }ooif ;  w  =  -  /, 


1 1 1 


Observed  on  penta-erythrite,  C5H12O4. 

TETRAGONAL  HBMIHEDRISMS. 

In  this  system  three  types  of  hemihedrisms 
are  possible,  figures  293,  294  and  295. 

a)  Sphenoidal  hemihedrism.  The  principal  an^tw -second- 
ary planes  divide  space  into  octants.  All  faces  in  alternate  octants 
are  suppressed,  the  others  expanded,  figure  293.'-,'  >' 


Fig.  292. 


Fig.  293. 


Fig.  294. 


Fig.  295. 


b)  Pyramidal  Hemihedrism.  The  two  secondary  and  the 
two  intermediate  planes  of  symmetry  divide  space  into  eight  equal 
sections,  figure  272.  All  faces  lying  wholly  within  alternate  sections 
are  extended,  the  others  suppressed,  figure  294. 

•c)  Trapezohedral  hemihedrism.  By  means  of  the  five  planes 
of  symmetry  sixteen  equal  sections  result.  In  this  type  of  hemihe- 
drism all  faces  lying  wholly  within  such  sections  are  alternately  ex- 
tended and  suppressed,  figure  295. 

The  pyramidal  and  sphenoidal  types  of  hemihedrism  are  the  most 
important. 


102 


TETRAGONAL    SYSTEM. 


3.     TETRAGONAL  SCALENOHEDRAL  CLASS.1) 

(Sphenoidal  Hemihedrism.} 

Symmetry.  In  this  class  the  elements 
of  symmetry  consist  of  two  intermediate 
planes  of  symmetry,  two  axes  of  binary 
symmetry  parallel  to  the  secondary  crystal- 
lographic  axes,  and  one  binary  axis  parallel 
to  the  principal  or  c  axis,  figure  296.  The 
sphenoidal  hemihedrism  is  analogous  to  the 
tetrahedral  and  rhombohedral  hemihedrisms 
of  the  cubic  and  hexagonal  systems,  respect- 
ively; see  pages  26  and  60. 

Tetragonal  bisphenoids.  Figure  298  shows  the  application  of 
the  sphenoidal  hemihedrism  to  the  bipyramid  of  the  first  order.  It 
is  obvious  from  this  figure  that  the  bipyramid  of  the  first  order  now 
yields!  two  i>ew  correlated  forms,  each  bounded  by  four  faces.  When 
the  development  is  ideal,  these  faces  are  equal  isosceles  triangles.  On 
account  of  the  resulting  forms  resembling  a  wedge,  they  are  termed  the 
-positive  and  negative  tetragonal  bisphenoids  of  the  first  order. 


Fig.  296. 


Fig.  297. 

The  symbols  are : 

f#  :  a  :  me 
~*~ 


Fig.  298. 


Fig.  299. 


;  + 


m? 


,  *\hhl\  figure  299; 


<m  p 

—  —  ,  K\hhl\  figure  297. 


l)    Sphenoidal  group  of  Dana. 


TETRAGONAL  SCALENOHEDRAL  CLASS. 


103 


The  secondary  crystallographic  axes  bisect  the  four  edges  of 
equal  length,  while  the  principal  axis  passes  through  the  centers  of 
the  other  two. 

Tetragonal  scalenohedrons.  It  is  readily  to  be  seen  from  figure 
301,  that  the  ditetragonal  bipyramid  also  yields  two  new  correlated 
forms.  These  consist  of  eight  similar  scalene  triangles,  and  are 
termed  the  positive  and  negative  tetragonal  scalenohedrons. 


Fig.  300. 


Fig.  301. 


Fig.  302. 


The  symbols  are: 

a  :  na  :  mc\ 


. 
;+ 


mPn 


mP  n 


\hkl\  figure  300. 


figure  302; 


The  principal  axis  joins  the  tetrahedral  angles,  which  possess 
two  pairs  of  equal  edges.  The  secondary  axes  bisect  the  four  zigzag 
edges. 

The  sphenoids  and  scalenohedrons  are  congruent  and,  hence, 
the  positive  and  negative  may  be  made  to  coincide  by  means  of  a 
revolution  through  90°  about  the  principal  axis. 

The  other  tetragonal  forms  are  apparently  holohedral  as  a  study 
of  figures  296,  298,  and  301  will  show. 

The  characteristics  of  the  forms  of  the  tetragonal  scalenohedral 
class  may  be  tabulated  as  follows: 


104 


TETRAGONAL    SYSTEM. 


FORMS 

SYMBOLS 

I 

Solid  Angles 

Tetrahedral 

Weiss 

Naumann 

Miller 

Trihedral 

Bisphenoids 
First  order 

,    f  a  :  a  \  me  ) 

,   mP 

K\hhl\ 

K\hhl\ 

: 

j 

4 

\ 

2 

mP 

2 

±{     *  ~J 

Bipyramid 
Second  order 

a  :  oca  :  me 

mPoo 

K{hol\ 

Apparently 
holohedral 

Tetragonal 
Scalenohedrons 

,    [a  :  na  :  mc\ 

f       mPn 

K\hkl\ 
K\h~kl\ 

I-  8 

j 

2+4 

1    - 

mPn 

2 

-(     *     J 

Prism 
First  order 

a  :  a  :  ooc 

OOP 

K\IIO} 

Apparently 
holohedral 

j 

Prism 
Second  order 

a  :  ooa  :  ooc 

OOPOO 

K\IOO\ 

Ditetragonal 
Prisms 

a  :  na  :  ooc 

oo  Pn 

K\hkO\ 

Basal  Pinacoid 

oca  :  oca  :  c 

OP 

K{OOI] 

Combinations.      Figure  303,    m    =   ooP,    {noj;  o  =  -\ — , 
1 1 1 1 ;  and  c  =  OP,  jooi  \ .     Urea,  CH4N2O. 


Fig.  303. 


Fig.  304. 


Fig.  305. 


TETRAGONAL    BIPYRAMIDAL    CLASS. 

Figures  304  to  307.     p  = 


105 


-,     in;  r  =  —  -, 


in  \- 


e=   Poo,  {ioij;   z  =  2Poc, 
|20i};  c  =  OP,    jooi  j;   m  — 


2P 
OOP,   \l\Q\\t   =  —— , 


22 


Fig.  306. 

u  =   P2,  5212}.     These  com- 
binations occur  on  chalcopyrite,  CuFeS2. 


Fig.  307. 


4.     TETRAGONAL  BIPYRAMIDAL  CLASS.1) 

(Pyramidal  Hemihedrism. ) 

Symmetry.  The  elements  of  this  class  are  one  principal  plane, 
a  center,  and  an  axis  of  tetragonal  symmetry,  as  shown  in  figure  312. 

Tetragonal  bipyramids  of  the  third  order.  From  figure  309  it 
is  obvious  that  by  means  of  the  pyramidal  hemihedrism,  the  dihex- 
agonal  bipyramid  yields  two  new  forms,  each  bounded  by  eight  equal 
isosceles  triangles,  figures  308  and  310,  termed  the  tetragonal  bipyr- 
amids of  the  third  order. 


The  symbols  are: 

(a  :  na  :  mc\ 


2 

mPn 
2 


310; 


I,  Tr\hkl\,  figure  308. 


In  form  these  bipyramids  do  not  differ  from  those  of  the  first  and 
second  orders.  Figures  3 1 1  and  3 1 3  show  the  positions  occupied  by 
the  bipyramids  of  the  three  orders  and  it  is  obvious  that  the  position 
of  the  forms  of  the  third  order  is  intermediate  between  that  of  the 
other  two. 

Tetragonal  prisms  of  the  third  order.  In  an  analogous  manner 
the  ditetragonal  prism,  figure  31  5,  yields  two  correlated  prisms  of  the 
third  order,  each  consisting  of  four  faces.  The  position  of  these 
prisms  in  respect  to  the  crystallographic  axes  is  illustrated  by  figures 
311  and  313. 


1 )     Dana's  pyramidal  group. 


106 


TETRAGONAL    SYSTEM. 


Fig.  314. 

The  symbols  are: 

,  a  :  na  :  ccc 


Fig.  315. 


[acPn  1 
— j-J,  'JAM.  figure  316. 

ko\,  figure  314. 


TETRAGONAL    BIPYRAMIDAL    CLASS. 


107 


Being  open  forms,  they  are  shown  in  combination  with  the  basal 
pinacoid. 

Other  forms.  The  other  holohedrons,  bipyramids  and  prisms  of 
the  first  and  second  orders  as  also  the  basal  pinacoid,  remain  unal- 
tered by  this  type  of  hemihedrism  because  their  faces  do  not  lie 
wholly  within  the  sections  formed  by  the  secondary  and  intermediate 
planes  of  symmetry,  compare  figures  309,  312,  and  315.  They  are, 
hence,  only  apparently  holohedral. 

The  chief  features  of  the  forms  of  the  tetragonal  bipyramidal 
class  are  given  in  the  following  table: 


FORMS 

SYMBOLS 

5 

rt 
fc, 

Solid  Angles 

Weiss 

Naumann 

Miller 

Tetrahedral 

Bipyramids 
First  order 

a  :  a  :  me 

mP 

v\hhl\ 

Apparently 
holohedral 

J 

Bipyramids 
Second  order 

a  :  aoa  :  me 

mPao 

*\hol\ 

Bipyramids 
Third  order 

,    \a\na  \  me] 

,  fmPn] 

*\hkl\ 

TT\h~kl\ 

1 
1-8 

J 

2  +  4 

L~J 

YmPn~\ 

-(             *             I 

i    L  *  J 

Prism 
First  order 

a  :  a  :  aoc 

OOP 

ir\IIO\ 

1 

1  Apparently 
1    holohedral 

J 

Prism 
Second  order 

a  :  Oca  :  Ode 

00  POO 

TT\IOO\ 

Prisms 
Third  order 

,    (a  :  na  :  occ] 

(+[*?] 

rooPwi 

-1—  J 

TT\hkO\ 

ie\hko\ 

U 
J 

—  I               2               J 

Basal  Pinacoid 

do  a  :  oca  :  c 

OP 

ir{oOI\ 

Apparently 
holohedral 

108 


Fig.  317. 


TETRAGONAL   SYSTEM. 

Combination.    Figure  317.    e  —  Poo,  TT|  101  \ ; 


This    combination    has    been 


-  I,     7^1315 
observed  on  the  mineral  scheelite,  CaWO4. 


5.     TETRAGONAL   TRAPEZOHEDRAL   CLASS.1) 

(  Trapezohedral  hemihedrism. ) 

Symmetry.  This  class  possesses  a 
principal  axis  of  tetragonal  symmetry  and 
four  axes  of  binary  symmetry.  Figure  3 1 8 
shows  these  elements  as  well  as  the  applica- 
tion of  the  trapezohedral  hemihedrism. 

Tetragonal  trapezohedrons.    The  only 
holohedron  which  can  be  affected  by  trap- 
Fig.  318.  ezohedral    hemihedrism,    page  101,    is    the 
ditetragonal    bipyramid,   figure  320.      This 

bipyramid  yields  two  new  correlated  forms,  bounded  by  eight  faces, 
which  are  similar  trapeziums,  figures  321  and  319.  Being  enantio- 
morphous,  they  are  termed  the  right  and  left  tetragonal  trapezo- 
hedrons. 


Fig.  319. 


Fig.  320. 


Fig.  321. 


Dana  rails  this  the  traprzohcdtal group. 


TETRAGONAL    TRAPEZOHEDRAL    CLASS. 


109 


The  symbols  are: 
_  fa  :  na  : 


,  .  _  7)     . 
,  r\hkl\,  figure  321; 


,  r\hki\t  figure  319. 


Alt  other  forms  are  apparently  holohedral  because  their  faces  do 
not  lie  wholly  within  each  of  the  sixteen  sections  resulting  from  the 
intersection  of  the  five  planes  of  symmetry. 

The  chief  features  of  the  forms  of  this  class  are  given  below: 


FORMS 

SYMBOLS 

en 

V 

u 

OC 

t, 

Solid  Angles 

Trihedral 

Tetrahedral 

Weiss 

Naumann 

Miller 

Bipyramids 
First  order 

a  :  a  :  me 

mP 

r\h7ll\ 

J 

Apparently 
'  holohedral 

Bipyramids 
Second  order 

a  :  oca  :  me 

mPoo 

r\hol\ 

Tetragonal 
Trapezohe- 
drons 

(a\na\  me} 

ri                       i 

i      mPn 

r\hkl\ 

T\h~kl\ 

1 

>8 

8 

2 

i'   - 

]     mPn 

[              2             J 

'    I     ' 

\             2 

Prism 
First  order 

a  :  a  :  ccc 

ocP 

r\\\Q\ 

>, 

Dparei 
holol 

fitly 
ledral 

Prism 
Second  order 

a  :  oca  :  occ 

oopoc 

T\  100  \ 

Ditetragonal 
Prism 

a  \  na  :  oo  c 

oo  Pn 

r\hko\ 

Basal  Pinacoid 

Oc  a  :  00  a  :  c 

OP 

r\OOI\ 

Thus  far,  the  tetragonal  trapezohedron  has  not  been  observed 
on  either  minerals  or  artificial  salts.  There  are,  however,  a  number 
of  compounds  like  sulphate  of  strychnine,  (C21H22N2O2)2H2SO4H-6H2O, 
and  sulphate  of  nickel,  NiSO4+6H2O,  which  have  been  referred  to 
this  class  of  symmetry  by  means  of  etch  figures. 


110 


TETRAGONAL    SYSTEM. 


6.     TETRAGONAL  PYRAMIDAL  CLASS.1) 

( Pyramidal  Hemihedrism  with  Hemimorphism. ) 

Symmetry.  If  the  forms  of  the  tetra- 
gonal bipyramidal  class,  page  1 50,  become 
hemimorphic,  the  principal  plane  and  center 
of  symmetry  disappear.  Hence,  the  only  ele- 
ment of  symmetry  of  this  class  is  a  polar  axis 
of  tetragonal  symmetry  which  is  parallel  to 
the  c  axis,  figure  322. 

Forms.  The  bipyramids  of  the  tetragonal 
bipyramidal  class  now  yield  upper  and  /oz^r  pyramids.  The  basal 
pinacoid  is  also  divided  into  two  forms.  The  prisms  remain  unaltered. 

Since  the  ditetragonal  bipyramid  is  now  divided  into  four  pyr- 
amids, this  class  is  sometimes  considered  a  tetartohedral  class. 

The  chief  features  of  the  forms  are  given  in  the  following  table: 


Fig.  322. 


FORMS 

SYMBOLS 

FACES 

Weiss 

Naumann 

Miller 

Pyramids 
First  order 

f  a  :  a  :  me  ]  « 

I  —T-  J  7 

f      «£. 

t  I 

\hhi\ 

j**7i 

}' 

Pyramids 
Second  order 

f  a  '.  QCi  :  me  ]  u 

1 

mPQC 

j*»'j 

5*<<( 

4 

-         " 
™/>QC    , 

[           2           J    / 

:               ^        7 

Pyramids 
Third  order 

_l_  f  a  :  na  :  mc^\  u 

• 

+  f^]   « 
+  ^J; 

-f2«!l. 

1      4     J 

-  [=^]  ' 

l"'j 
{**/! 

l*i,  j 

i  **'~S 

* 

~[          4          J    / 

Prism 
First  order 

a  :  a  :  QCt 

' 

-  Apparently  of  tetragonal  bipyramidal  class 

Prism 
Second  order 

a  :  QCa  :  Q£c 

Prisms 
Third  order 

i    f  a  :  na  :  QCc  ] 

-  I            2           J 

Basal  Pinacoids 

f  QCa  :  Ota  :  c  ]  u 

• 

OP 
u 

k< 

\°°'\ 

!• 

1                    2                    J      / 

{•"I 

The pyramidal-hemimorphic group  of  Dana. 


TETRAGONAL    BISPHENOIDAL    CLASS. 


Ill 


Combination.     The    mineral    wulfenite, 
PbMoO4,   crystallizes  in  this  class,  figure  323. 

;  p  = 


P 

o  =  - 


in 


P 

o'     ~-  -    /, 


7.     TETRAGONAL  BISPHENOIDAL  CLASSY 

(  Tetartohedrism. ) 

Symmetry.  The  only  element  of  sym- 
metry remaining  in  this  class  is  an  axis  of 
binary  symmetry  parallel  to  the  c  axis,  figure 
324- 

Tetartohedrism.  The  forms  of  this  class 
may  be  conceived  as  derived  from  the  holo- 
hedrons  by  the  simultaneous  occurrence  of 
either  the  sphenoidal  and  trapezohedral,  or 
the  sphenoidal  and  pyramidal  hemihedrisms; 
compare  figures  293,  294,  and  295.  The 
simultaneous  occurrence,  however,  of  the 
pyramidal  and  trapezohedral  hemihedrisms 
give  rise  to  the  forms  as  described  in  the  pre- 
ceding class. 

Hi  sphenoids  of  the  first  order.  Figure 
325  shows  that  the  bipyramid  of  the  first  order 
now  yields  two  bisphenoids  of  the  same  order. 
Compare  figure  298. 

The  symbols  are: 


Fig.  325. 


Bisphenoids  of  the  second  order.     The  bipyramid  of  the  second 
order,  figure  327,  gives  rise  to  the  positive  and  negative  bisphenoids 


Dana  terms  this  the  tetartohedral group. 


112 


TETRAGONAL   SYSTEM. 


of  the  second  order.     These  bisphenoids  bear  the  same  relation  to 
those  of  the  first  order  as  do  the  corresponding  bipyramids. 


Fig.  326. 

The  symbols  are: 
,    ( a  :  oca  :  mc\ 


ohl\y  figure  326. 


Fig  327. 


Fig.  328. 


mPoo  .                                      mPoo 
—j—\hol\,  figure  328; — 


Fig.  329. 


Fig.  330. 


Fig.  331. 


Fig.  332. 


Fig.  333. 


Fig.  334. 


TETRAGONAL   BISPHENOIDAL   CLASS. 


113 


.Bisphenoids  of  the  third  order.  In  a  like  manner,  as  is  obvious, 
the  ditetragonal  bipyramid,  figure  330,  yields  four  bisphenoids  of 
the  third  order.  Figure  333  shows  that  these  forms  may  also  result 
when  hemihedrism  is  applied  to  the  tetragonal  scalenohedron. 

The  symbols  are: 

[a  :  na  :  me]  mPn    .  _ 

Positive  right,   +  r\ "•+*"—    — »  \khl\^  figure  321. 


Positive  left,   -f  / 


Negative  right,  —  r 


a  :  na  :  me 


,   +/ 


a  :  na  :  me 


4 
mPn 


\hkl\,  figure  329. 
,  \khl\,  figure  334. 


Negative  left,   —  / 


fa  : 


na  :  me 


/-    -,\hkl\,  figure  332. 


The  same  relationship  exists  between  the  bisphenoids  of  the 
three  orders  which  was  noted  with  the  corresponding  prisms,  page 
105. 

In  bisphenoids  of  the  first 
order  the  secondary  axes  bisect 
the  zigzag  edges.  They  join 
the  centers  of  opposite  faces 
in  forms  of  the  second  order, 
while  in  those  of  the  third  order 
they  occupy  an  intermediate 
position. 


Fig.  335. 

Prisms  of  the  third  order.  Figure  335  shows  that  the  ditetra- 
gonal prism  now  yields  two  prisms  of  the  third  order,  compare 
figures  314  and  316. 


OoPn  . 


The  symbols  are: 

(  a:  na  :  doe] 
-{  2  j  ' 

The  other  forms,  prisms  of  the  first  and  second  orders  and  the 
basal  pinacoid,  remain  unchanged. 


114  TETRAGONAL   SYSTEM. 

The  principal  features  may  be  tabulated  as  follows: 


FORMS 

SYMBOLS 

1 

Trihedral 
Solid 
Angles 

Weiss 

Naumann 

Miller 

Bisphenoids 
First  order 

+ 

[a  :  a  :  mc~\ 

- 

mP 

\hhi\  } 

-4 
\hhl\ 

4 

2 

mP 

2 

\           *            \ 

Bisphenoids 
Second  order 

±_ 

a:  ooa:  me  1 

\ 

\ 

mPoc 

\hol\ 
\ohl\ 

1 

u 

J 

4 

2 

mPoo 

2           J 

2 

V 

Bisphenoids 
Third  order 

r  [  a  :  na  :  mc\ 

-< 

mPn 

\kht\ 
\hhl\ 
\hkl\ 
\h~kl\ 

1 

-4 

> 

4 

fr     4 
mPn 

—  T 

+  l^n 

l  mPn 

±-/i"     4         J 

l 

^            4 

Prism 
First  order 

a  :  a  :  ooc 

Appa 

rently  h 

oloh 

edral 

Prism 
Second  order 

a  :  ooa  :  ooc 

Prisms 
Third  order 

±1 

a  :  na  :  00^:1 

] 

' 

,  mPoo 

\hko\ 
\h~ko\ 

] 

i4 

2 

mPtt 

2               J 

2 

Pasal  Pinacoid 

ooa  :  ooa  :  c 

Apparently  holohedral 

representatives   of   this   class  of   symmetry  have  yet   been 


No 
observed, 


-C 


ORTHORHOMBIC  SYSTEMS 

Crystallographic  Axes.    This  system  includes  all  forms  which 
can  be  referred  to  three  unequal  and  perpendicular  axes,   figure  336. 
One  axis   is   held  vertically,    which   is,    as 
heretofore,  the  c  axis.     Another  is  directed  ^ 

toward  the  observer  and  is  the  a  axis,  some- 
times    also    called    the    brachy  diagonal. 

The  third    axis  extends   from  right  to  left      ~» — 

and    is    the    b    axis    or     macrodiag'onal. 

There    is  no  principal  axis  in    this  system, 

hence  any  axis  may  be  chosen  as  the  vertical  Fig.  336. 

or   c  axis.      On  this  account  one  and  the 

same    crystal    may     be    held     in    different    positions     by     various 

observers,     which     has     in    some    instances    led     to    considerable 

confusion,    for,    as    is    obvious,    the    nomenclature    of    the    various 

forms    cannot    then    remain    constant.       In    this    system    the    axial 

ratio  consists  of    two  unknown    values,  viz:    a\b  :  c  =  .8130:  1  : 

I-9°37»   compare   page   7. 

Classes  of  Symmetry.     The    orthorhombic    system    comprises 
three  classes  of  symmetry,  as  follows: 

i.     Orthorhombic  bipyramidal  class  (Holohedrism.) 

Holohedrism  and  1 


2.  Orthorhombic  pyramidal  class  ,        .         ... 

[  hemimorphism. 

3.  Orthorhombic  bisphenoidal  class         (Hemihedrism.) 

Numerous  representatives  of  all  these  classes  have  been  observed 
among  minerals  and  artificial  salts.  The  first  class  is,  however,  the 
most  important. 

/.     ORTHORHOMBIC  BIPYRAMIDAL  CLASS.*) 

(Holohedrism.} 

Symmetry,  a)  Planes.  There  are  three  planes  of  symmetry. 
These  are  perpendicular  to  each  other  and  their  intersection  gives  rise 


!)    Sometimes  termed  the  trimetric,  rhombic,  or  prismatic  system. 
2)    The  normal  group  ol  Dana. 


116 


ORTHORHOMBIC   SYSTEM. 


Fig.  337. 


to  the  crystallographic  axes,  figure  337. 
Inasmuch  as  these  planes  are  all  dissimilar, 
they  may  be  written: 

1  +  1  +  1=3  planes. 

b)  Axes.     Three  axes  of  binary  sym- 
metry are  to  be  observed,  figure  337.    They 
are  parallel  to  the  crystallographic  axes  and 
indicated  thus: 

!•+!•+!•=  3  axes. 

c)  Center.      This    element    of    sym- 
metry is  also  present  and  demands  parallel- 
ism of  faces.      Figure  338  shows  the  above 
elements  of  symmetry. 

Orthorhombic  bi  pyramids.    The  form 

whose  faces  possess  the  parametral  ratio,  a  :  b  :  c,  is  known  as  the 
unit  or  fundamental  orthorhombic  bipyramid.  It  consists  of 
eight  similar  scalene  triangles,  figure  339.  The  symbols  according 
to  Naumann  and  Miller  are  as  follows: 

P,   {nij. 


Fig.  338. 


Fig.  339. 


Fig.  340. 


Fig.  341. 


The  outer  form,  in  figure  340,  possesses  the  ratio  a  :  b  :  me 
(in  >  o  <  oc).  In  this  case  m  =  2.  This  is  a  modified  orthorhom- 
bic bipyramid.  Its  symbols  are : 

mP,  \hhl\. 

In  figure  341,  the  heavy,  inner  form  is  the  unit  bipyramid.  The 
lighter  bipyramids  intercept  the  b  and  c  axes  at  unit  distances  but  the 


ORTHORHOMBIC    BIPYRAMIDAL    CLASS. 


117 


a  axis  at  distances  greater  than  unity.  Their  ratios  may,  however, 
be  indicated  in  general  as, 

na  :  b  :  me,  (n  >  I ;  m  >  o  <  00 ). 

These  are  the  brachybipyramids,  because  the  intercepts  along  the 
br  achy  diagonal  are  greater  than  unity.'  The  Naumann  and  Miller 
symbols  are:  mPn,  \hkl\. 


Fig.  342. 


Fig.  343. 


Figure  342  shows  two  bipyramids  (outer)  which  cut  the  a  axis  at 
unity  but  intercept  the  b  axis  at  the  general  distance  nb,  (n  >  i). 
The  ratios  would,  therefore,  be  expressed  by  a  :  nb  :  me.  Since  the 
intercepts  along  the  macrodiagonal  are  greater  than  unity,  these  are 
called  macrobipyramids*  The  symbols  according  to  Naumann  and 
Miller  are  mPn,  \khl\. 

Figure  343  shows  the 
relationship  existing  be- 
tween the  unit,  macro-,  and 
brachy  bipyramids,  while 
figure  344  shows  it  for  the 
unit,  modified,  and  macro- 
bipyramids. 

Prisms.  Similarly 
there  are  three  types  of 
prisms,  namely,  the  unit, 

macro-,  and  brachyprisms .  Fig.  344. 

Each  consists  of  four  faces, 
cutting  the  a  and  b  axes,  but  extending  parallel  to  the  c  axis 


118 


ORTHORHOMBIC    SYSTEM. 


Figure  345  repre- 
sents a  tmit-prism  with 
the  following  symbols: 

a  \b  :  act,    ooP,    {no}. 

The  brachyprism  is 
shown  in  figure  346.  Its 
symbols  are: 

na\b\  oo^, 
\hko\ 


In   figure    347,    there 
is  a  unit  prism  surrounded 
by  a  macroprism,    whose   symbols 
may  be  written: 

a  :  nb  :  ccc,  ocPn,  \kho\. 

For  the  relationship  existing 
between  these  three  prisms  compare 
figure  343. 

Domes.     These  are  horizontal 

Fig.  347.  prisms  and,   hence,  cut  the  £  and 

one  of  the  horizontal  axes.     Domes, 

which  are  parallel  to  the  a  axis  or  brachydiagonal,  are  called  brachy- 
domes.     Their  general  symbols  are: 

oca  :  b  :  me,  mP$o,  \ohl\,  figure  348. 


Those,  which 
extend  parallel  to 
the  macrodiag- 
onal,  are  termed 
macro  domes, 
figure  349.  Their 
symbols  are: 


Fig.  348. 


Fig.  349. 


\hol\.      . 

As  is  obvious, 
prisms  and  domes 


ORTHORHOMBIC    BIPYRAMIDAL    CLASS. 


119 


are  open  forms  and,  hence,  can  only  occur  in  combination  with  other 
forms. 


Pinacoids.  These  cut  one  axis  and 
extend  parallel  to  the  other  two.  There 
are  three  types,  as  follows: 

Basal  pinacoids,  ooa:  Oo5  :  c,  OP, 

{001  j. 

Brachypinacoids,  <x«  •  5  :  00^,  OoPoo, 

JoioJ. 
Macropinacoids,    a  :  oob  :  ooe,   ooPoo, 

jiool. 


^*^'             ' 

^^ 

1        1 

1 

i       i 

!       i_ 

.-—  . 

I     "\ 
\       \ 

\       \ 

JL.        1 

^^'                        I 

^ 

Fig.  350. 


These  forms  consist  of  two  faces. 
Figure  350  shows  a  combination  of  three 
types  of  the  pinacoids. 

The  characteristics  of  the  forms  of  this  class  are  given  in  the 
following  table: 


FORMS 

SYMBOLS 

1 

Tetrahedral 
Solid 
Angles 

Weiss 

Naumann 

Miller 

Orthorhombic 
Bipyramids 

Unit 

a:  b'.c 

P 

I///] 

-8 
4 

2  +  2+2 

Modified 

a  :  b  :  me 

mP 

\hhi\ 

Brachy 

na  :  b  :  me 

mPH 

\hki\ 

Macro 

a  \nb  :  me 

mPn 

\khi\ 

Orthorhombic 
Prisms 

rUnit 

a  \b  :00r 

OOP 

\IIO\ 

Brachy 

na  :  b  :  00  c 

OoPn 

\hko\ 

LMacro 

a  :  nb  :  ace 

aoPTi 

\kho\ 

Domes 

'  Brachy 

ooa  :  b  :  me 

mPao 

\ohl\ 

1- 

Macro 

a:  00  b  :  me 

mPoo 

\hol\ 

Pinacoids 

r  Basal 

ooa  :  OOb  :c 

OP 

{001} 

F 

Brachy 

aoa  :  b  :  doe 

ooPfc 

\oio} 

Macro 

a  :  dob  :  OOe 

ooPao 

\IOO\ 

120 


ORTHORHOMBIC   SYSTEM. 


Relationship  of  forms.    This  is  to  be  seen  from  the  following 
diagram : 


ooPoo 


ooPn 


OOP 


oo  Pn 


ocPoo 


Fig.  351. 


Fig.  352. 


Fig.  353. 


Fig.  354. 


Combinations.  Figure  351.  ^=ooP,  {no};  e  =  P2,  {122}. 
Brookite,  TiO2. 

Figure  352.  c  =  OP,  {ooi};  *  =  iP,  {113};  e  =  fPoc,  Jo23}. 
Chalcocite,  Cu2S. 

Figure  353.  P  =  P,  fin};  s  =  lP,  ju3};  c=  OP,  \ooi\. 
Sulphur,  S. 

Figure  354.  m=ooP,  }iio}.;  b  =  ooPoo,  }oio};  k  =  Poo, 
ion}.  Aragonite,  CaCO8. 

Figure  355.  a  =  OoPoo,  j  100} ;  5  =  OoP2,  1 120} ;  b  =  OcPoo, 
{oio};  o  =  P,  jiii};  ^  =  2P2,  \2ii\-  t  =  PS)t\on}.  Chryso- 
beryl,  Be(AlO2)2. 


ORTHORHOMBIC    PYRAMIDAL    CLASS. 


121 


Fig.  355. 


Fig.  356. 


Fig.  357. 


Figure  356.  m  =  ooP,  \  i  io\ ;  /  =  ooP2, 
JI20J;  ^  =  2Pdo,  {021};  o  =  P,  jiuf.  Topaz, 
Al2(F.OH)2Si04. 

Figures  357  to  359.  m  =  OoP,  SI][o!; 
w  =  OoP|,  J320jj  0  =  ocPoc,  j  loof ;  z  =  P, 
{in};  17  ==  fPf,  {324};  <:  =  OP,  Sooi};  o  = 
OCPO),  |ouj;  rf  =  |Poo,  |i02|;  x  =  ?4, 
{414};  j  =  P2  JI22}.  Celestite,  SrSO4. 


Fig.  358. 


Fig.  359. 


2.     ORTHORHOMBIC  PYRAMIDAL  CLASS.1) 

(Holohedrism   with  Hemimorphism.} 

Symmetry.      As  figure  360  shows,   the  elements  of  symmetry  of 
this  class  are  the  vertical  planes  of  symmetry  and  one  axis  of  binary 


Fig.  360. 

symmetry  parallel  to  the  c  axis.     This  axis  of  symmetry  is  obviously 
polar. 

All  forms  with  the  exception  of  the  prisms,  the  macro-  and 
brachypinacoids  are  now  divided  into  upper  and  lower  forms,  as 
shown  in  the  following  tabulation: 


1)    The  hemimorphic  group  of  Dana. 


122 


ORTHORHOMBIC    SYSTEM. 


FORMS 

SYMBOLS 

Faces 

Tetra- 
hedral 
Solid 
Angles 

Weiss 

Naumann 

Miller 

Ortho- 
rhombic-! 
Pyramids 

Unit 

\a\b  \c^u 
[2       J  / 

P 

—  u 

£ 

\III\ 

\II~I\ 

"I 

-  4 

J 

I 

Modified 

\a  \b  :  me]  u 

mP 
u 

'  ^ 

\hhl\ 
\hh~l\ 

{     ^     \i 

Brachy 

\na  :  b  :  me]  u 

mPn 

{hkl} 

\hkl\ 

u 
mPn 

(     *     \i 

I        2 

Macro 

. 

\a  :  rib  :  me]  u 

mPn 

\khl\ 
\khl\ 

u 
\  mPTi  i 

I              2               }l 

\          *       [ 

Ortho- 
rhombic^ 
Prisms 

Unit 

a  :  b  :  CCc 

OOP 

\IIO\ 

Appar- 
j_        ently 
holo- 
he  dral 

Brachy 

na  :  b  :  ooc 

00  Pn 

\hko\ 

Macro 

a  :  nb  :  ace 

aoPTi 

\kho\ 

Domes       •* 

' 
Brachy 

{  oca  :  b  :  me]  u 

(mP<x> 
11 

\ohl\ 
\ohJ\ 

2 



\    * 

\  mPac  7 

[                 2               \l 

1         2        l 

Macro 

\a  :  oo5  :  me]  u 

(mPoo 

Ju 

\hol\ 
\hol\ 

*r 

j  mPoo  r 

[              2               \l 

(        2         l 

Pinacoids 

Basal 

f  ooa  :  QOb  :c]  u 

r    OP 
u 

\  °pl 

\OOI\ 
\OOI\ 

1 
•     I 
J 

[      2     -J? 

1         2     l 

Brachy 

ooa  :  b  :  ooc 

ooP66 

\OIO\ 

\ 
Apparently 

\  holohedral 

Macro 

a  :  oob  :  ooc 

oo  Poo 

\IOO\ 

ORTHORHOMBIC    BISPHENOIDAL    CLASS. 


123 


Combinations.     Figure  361.      a   =  ooPoo,   |ioo};    g-  =  ooP, 


}noj;    b  =ooPo6,  joio};    o  = 


3  Poo 


101 


ut  J3Oi}; 


;  r  =  -u,    ou    ;    <?  = 


,  {ooij;   5  = 


ji2ij.      Calamine,  also  called  hemimorphite,  Zn2(OH)2SiO3. 


Fig.  362. 


Figure  362. 


=  -   -  w,    j  101  1  ;    r  =  -    -  w,    joi  i  j  ; 

2  2 


{ool}.     Struvite,  NH4MgPO4-f  6H2O. 


3.     ORTHORHOMBIC  BISPHENOIDAL  CLASS.1) 

( Sphenoidal  Hemihedrism. ) 

Symmetry.  As  can  be  seen  from  figure  363,  this  class  possesses 
three  axes  of  binary  symmetry.  The  other  elements  have  disappeared. 
The  method  of  extension  and  suppression  of  faces  is  comparable  to 
the  tetrahedral  and  sphenoidal  hemihedrisms  of  the  cubic  and  tetra- 


1)     Dana  terms  this  class  the  sphenoidal  group. 


124 


ORTHORHOMIC   SYSTEM. 


gonal  systems,  respectively,  in  that  faces  lying  wholly  within  alternate 
octants  are  extended. 


Orthorhombic  Msphenoids.  Figure  365  shows  the  application 
of  the  sphenoidal  hemihedrism  to  an  orthorhornbic  bipyramid,  which 
yields  two  new  correlated,  enantiomorphous  forms,  called  the  right, 
figure  366,  and  left,  figure  364,  orthorhornbic  bisphenoids.  These 
forms  are  bounded  by  four  scalene  triangles.  The  crystallographic 
axes  bisect  the  edges. 


Fig.  364. 


Fig.  365. 


Fig..  366. 


Since  there  are  several  types  of  orthorhombic  bipyramids,  it  fol- 
lows that  each  will  yield  two  enantiomorphous  bisphenoids.  They 
are  termed  the  orthorhombic  bisphenoids.  These  are  given  and 
their  symbols  indicated  in  the  tabulation  on  page  125. 

All  other  forms  must  occur  with  the  full  number  of  faces,  that  is, 
they  are  apparently  holohedral. 


ORTHORHOMBIC    BISPHENOIDAL   CLASS. 


125 


FORMS 

SYMBOLS 

7aces 

Trihe- 
dral 
Solid 
Angles 

Weiss 

Naumann 

Miller 

Ortho- 
rhombic 
Bisphe- 
noids 

Unit 

(S:  5:<H 

1        P 
r~J 

n 

\ni\ 

\i7i\ 

4 

4 

'F/[          *          I 

Modified 

,(a:5  :mf: 

"j       mP 

\hhl\ 
\hhl\ 

\j± 

r,*[        2 

Brachy 

.  f  na  :  b  :  me 

V       I 

ImPn 
mPh 

\hkl\ 
\h~kl\ 

r'l{    2 

1     2 

Macro 

. 

,  (a  :  nB  :  m£\ 

]       mPn 

\khl\ 
\khl\ 

V  * 

''        I              2              J 

mPn 

J      l       * 

Ortho- 
rhombicH 
Prisms 

Unit 

a  \l  :ooc 

OOP 

\!,0\ 

Apparently 
holohedral 

Brachy 

na  :b:Qo£ 

aoPn 

\hko\ 

Macro 

a  \nb  :  OCc 

ccPn 

\kho\ 

Domes 

Brachy 

oo  a  :  b  :  md 

mPoo 

\ohl\ 

Macro 

a  :  Oo5  :  mt 

mPtt 

\hol\ 

Pinacoids 

Basal 

oca  :  Oc5  :  d 

OP 

\OOI\ 

Brachy 

oca  :  b  :  oc<^ 

oo  Pob 

\OTO\ 

^Macro 

a  :  aob  :  00^ 

oo  Poo 

\IOO\ 

Combination.     Figure    367.     m   =  ooP,     {noj; 

P       -  P 

/  — ,  { 1 1 1 } ;  z  =  r  — ,  1 1 1 1 } .     Epsomite,  MgSO4  + 

2  2 


Fig.  367. 


MONOCLINIC  SYSTEMS 

Crystallographic  Axes.     To  this   system   belong   those   forms 
which  can  be  referred  to  three  unequal  axes,  two  of  which  (a'  and  c) 

intersect  at  an  oblique  angle,  while  the 
third  axis  (b)  is  perpendicular  to  -these 
two.  The  oblique  angle  between  the  a' 
and  c  axes  is  termed  ft.  Figure  368 
shows  an  axial  cross  of  this  system. 


-a. 


p I*/  T  It  is  customary  to  place  the  b  axis  so 

as  to  extend  from  right  to    left.     The  £ 
+€tf  axis  is  held  vertically.     The  a'  axis  is  then 

directed  toward  the  observer.     Since  the 
a'   axis  is  inclined,  it  is  called  the  clino- 
axis.     The  b  axis  is  often  spoken  of  as 
Fig  368.  the  ortho-axis.     The  obtuse  angle  be- 

tween the  a'  and  c  axis  is  the  negative 

angle  ft,  whereas  the  acute  angle  is  positive.  Obviously,  they  are 
supplementary  angles.  The  elements  of  crystallization  consist  of  the 
axial  ratio  and  the  angle  ft,  which  may  be  either  the  obtuse  or  the 
acute  angle.  Compare  page  8. 

Classes  of  Symmetry.     The  monoclinic  system  includes  three 
classes  of  symmetry,  as  follows: 

1.  Prismatic  class         (Holohedrism^) 

2.  Domatic  class          (Hemihedrism.) 

3.  Sphenoidal  class    (Hemimorphism.) 

The  first  class  is  the  most  important. 

/.     MONOCLINIC  PRISMATIC  CLASS.2) 
(Holohedrism.} 

Symmetry.     This  class  possesses  one  plane  of  symmetry,  which 
includes  the  a'  and  £  axes  and,  hence,  is  directed  toward  the  observer. 


i)    Also  termed  the  clinorhombic,  hemiprismatic,  monoelinohedral,  monosymmetric  or  oblique 
system. 

3)    Normal  group  of  Dana.  [126] 


MONOCLINIC    PRISMATIC   CLASS. 


127 


Perpendicular  to  this  plane,  that  is,  parallel 
to  the  b  axis,  is  an  axis  of  binary  symmetry. 
A  center  of  symmetry  is  also  present. 
Figure  369  shows  these  elements  in  a  crystal 
of  augite.  These  elements  are  represented 
diagrammatically  in  figure  370,  which  is  a 
projection  of  a  monoclinic  form  upon  the 
plane  of  the  a'  and  b  axes. 

Hemi-pyramids.  On  account  of  the 
presence  in  this  class  of  only  one  plane  of 
symmetry  and  an  axis  of  binary  symmetry, 
a  form  with  unit  intercepts,  that  is,  with 
the  parametral  ratio  a'  :  b  :  c,  can  possess 
but  four  faces.  Figure  371  shows  four 
such  faces,  which  enclose  the  positive  angle 
ft  and  are  said  to  constitute  the  -positive 
unit  hemi-pyramid.^  Figure  372  shows 
four  faces  with  the  same  ratio  enclosing  the 


Fig.  369. 


Fig.  370. 


Fig.  371. 


Fig.  372. 


Fig.  373. 


negative  angle  /?,  and  comprising  the  negative  unit  hemi-pyramid. 
It  is  obvious  that  the  faces  of  these  hemi-pyramids  are  dissimilar, 
those  over  the  negative  angle  being  the  larger.  The  symbols  for  these 
hemi-pyramids  are  according  to  Naumann  and  Miller,  -HP  j  iilj  and 
—  P  1 1 1 1  { ,  respectively.  Two  unit  hemi-pyramids  occurring  simul- 
taneously constitute  the  monoclinic  unit  bipyramid,  figure  373. 


i)    The  term  hemi-bipyramid  ought  more  properly  be  employed  because  of  the  fact  that  we  are 
dealing  with  four  faces  of  a  bipyramid  and  not  of  a  pyramid  as  the  term  has  hitherto  been  used. 


128 


MONOCLINIC    SYSTEM. 


Since  this  system  differs  essentially  from  the  orthorhombic  in  the 
obliquity  of  the  a  axis,  it  follows  that  modified,  clino,  and  ortho 
hemi-pyramids  are  also  possible.  They  possess  the  following  general 
symbols: 

Modified  hemi-pyramids, 

±_(a'  :  b  :  me),  w>o 
Clino  hemi-pyramids, 

±(na'  :  b  :  me),  n>\; 
Ortho  hemi-pyramids, 

:nb  \mc\  n>i;    +  nt?7i  [hk7\\   -  -mVTi  \hkl\\  h>k. 


m?n'  \hkl\\— 


hkl\,  h<k. 


Fig.  374. 


Fig.  375. 


Prisms.  As  was  the  case  in  the 
orthorhombic  system,  page  1  17,  there  are 
also  three  types  of  prisms  possible  in  this 
system,  namely,  unit,  clino-,  and  ortho- 
prisms.  These  forms  cut  the  «'  and  b 
axes  and  extend  parallel  to  the  vertical 
axis. 

The  general  symbols  are: 

Unit  prism, 
a'  :  b  :  ace,  ooP,  j  iiof,  figure  374. 

Clinoprism, 

na'  :  b  :0cc,  ooPn',  \hko\\  n>i, 
h<k. 

Orthoprism, 

a'  :  nb  :  oc  c,  oc  P7I,  j  h  k  o  \  ;    n  >  i  , 
h>k. 

Domes.  In  this  system  two  types 
of  domes  are  also  possible,  namely,  those 
which  extend  parallel  to  the  a'  and  b  axes, 
respectively.  Those,  which  are  parallel 
to  a\  are  termed  clinodomes  and  consist 
of  four  faces,  figure  375.  The  general 
symbols  are: 

oca'  :  b  :  me,  mPcc',  \ohl\. 

Since  the  a'  axis  is  inclined  to  the 
c,  it  follows  that  the  domes  which  are 


MONOCLINIC    PRISMATIC    CLASS. 


129 


parallel  to  the  b  axis  consist  of  but  two  faces.  Figure  376  shows 
such  faces  enclosing  the  positive  angle  and  are  termed  the  positive 
hemi-orthodomc,  whereas  in  377  the  negative  hemi-orthodome  is 


Fig.  376. 


Fig.  31 


Fig.  378. 


represented.  It  is  evident  that  the  faces  of  the  positive  form  are 
always  the  smaller.  Figure  378  shows  these  hemidomes  in  com- 
bination. Their  general  symbols  are: 

Positive  hemi-orthodome, 

a'  :  ccZ>  :  me,  +  ;wPoc,  \hol\. 
Negative  hem i-orth odome, 

a'  :  GC&  :  m£,  — mPtt,  \hol\. 

Pinacoids,     There  are  three  types  of  pinacoids  possible  in  the 
monoclinic  system,  namely, 

Basal  pinacoids,  oca'  :  ccb  :  c, 
OP,  jooi  \. 

Clinopinacoids,      oca'    :  b    :    00 r, 
ooPoo',  joio}. 

Orthopinacoids,     a'    :   ccb    :   occ, 

OOPOC,    \100\. 

These  are  forms  consisting  of  but 
two  faces.  Figure  379  shows  a  combi- 
nation of  these  pinacoids. 

All  forms  of  the  monoclinic  system 
are  open  forms  and,  hence,  every  crystal 
of  this  system  is  a  combination. 


Fig.  379. 
A  summary  of  the  forms  of  this  class  is  given  as  follows: 


130 


MONOCLINIC   SYSTEM. 


FORMS 

SYMBOLS 

I 

s. 

Weiss 

Naumann 

Mi  ler 

Hemi- 

pyramids 

Unit 

±  (a'  :  b  :  c) 

(           +J> 
I          ~P 

[//}} 
\III\ 

' 
. 

^4 

^4 

Modified 

±(af  :b  :mc) 

j     -h  mP 
\     —mP 

\hhl\ 
\hhl\ 

Clino- 

±(na'  :  b  :  me) 

j   +mPn' 
(   —mPri 

\hkl\ 

\hhl  \ 

Ortho- 

±_(a'  :nb\mc) 

5    +mPn 

\     -  mPn 

\khl\ 
\khl\ 

Prisms 

Unit 

a'  :5:ooc 

OOP 

\I,0\ 

Clino- 

no!  :  b  :  ooc 

aoPn' 

\hko\ 

Ortho- 

a'  :  nb  :  ooc 

ooPn 

\kho\ 

Clinodome 

ooa'  :b  :  me 

mPoo' 

\ohl\ 

4 

Hemi- 
orthodomes 

Positive 

a'  :  oob  :  me 

+  mPo5 

\hol\ 

}> 

Negative 

a'  :  oob  :  me 

-mPoo 

\hol\ 

Pinacoids 

Basal 

ooa'  :  oob  \£ 

OP 

\OOI\ 

2 

Clino- 

aoaf  :  b  :  voc 

00  POO' 

\OIO\ 

Ortho- 

a'  :  00^  :  ooc 

oo  Poo 

\IOO\ 

Relationship 

of  forms. 

This  is  clearly  shown  by 

the  following 

diagram  : 

OP 

OP 

OP                   OP 

OP 

1 
i 

1 

1                        1 
-4-  l  P             +  '  PTi 

1 

m                   —  m 

\            \ 

Pm"                     -1-  P*»' 

~r       r               ~r      f)l 
—  m                —  m 

\             \ 

A-  P                     -1-  P7} 

'  1 
—  +  Poo 

"  i 

1              "1 

1      A       •                                   1      i   tl 

"1 

1 

r^PryV                  -4- 

i 

r^  P*? 

"1   _ 

rmPr/S 

MONOCLINIC    PRISMATIC    CLASS. 


131 


a 


TTl 


Fig.  381. 


Fig.  382. 


Fig.  383. 


Combinations.  Figure  380.  m  =ooP,  jnof;  b=  ooPoo', 
Joio};  p=—P,  jmf.  Gypsum,  CaSO4.2H2O. 

Figure  381.  c  =  OP,  |ooj|;  m  —  ooP,  jno|;  £=ooPoo', 
{010} ;  j  =  2Pob,  J20i{.  Orthoclase,  KAlSi3O8. 

Figure  382.  m=aoP,  juof;  <?  =  OP,  Sooi};  «  =  ooPoc, 
|iooj;  ^  =  Pooy,  {on}.  Thorium  sulphate,  Th(SO4)3.9H2O. 

Figure  383.  c  =  OP,  jooi};  a  =  ooPoo,  jioo};  b=aoPoo', 
;  w=  -P,  {iu|.  Diopside,  CaMg(SiO3)2. 


Fig.  384. 


Fig.  385. 


Fig.  386. 


Figure  384.  w=ooP,  {no};  ^  =  ooPoo',  {oio};  ^  =  OP, 
{ooij;  n  —  2Poo',  {021};  _y  =  2Poc,  {201};  o  =  P,  {iiij; 
jc1  =  Poo,  { 101 }.  Orthoclase. 

Figure  385.  a  =  ocPoo,  { ioo( ;  Z>  =  ooPoo',  {010} ;  u  =  — P, 
{in};  c=OP,  \ooi\-  m  =  ooP,  {no};  /  =oo?3,  {310}. 
Diopside. 

Figure  386.  c  —  OP,  jooi};  a  =  ocPoo,  {iooj;  r  =  Poo, 
jioi};  o  =  —  P,  {111};  »  =  +P,  {in};  g  =  -hPoo,  { ioT( ; 


132  MONOCLINIC    SYSTEM. 


{201  J  ;    q  ----  Poc'f  jouS  ;    TT  ==  2P2,  J2if};    I  = 
S  3  1  1  }  I        =  P3,  I  3  i  3  j  ;  *  =  —  3P3,  i  3  1  1  !  •    Praseodymium  sulphate, 
Pr2(S04)8.8H80. 

2.     MONOCLINIC  DOMATIC  CLASS.1) 

(  Hemihedrism.  ) 


^ 
^  —  -*-_ 


Symmetry.  This  class  possesses  but  one 
element  of  symmetry,  namelv,  a  plane  of 
symmetry  as  shown  in  figure  387. 

Tetra-pyramids.       Since     the     axis     of 

Fig" 387.  binary  symmetry  is  lost,   it  follows   that    the 

hemi-pyramids  now  yield  two  tetra-pyramids. 

These  forms  consist  of  but  two  faces  which  are  situated  symmetrically 
to  the  plane  of  symmetry.  For  example,  in  figure  371  the  positive 
hemi-pyramid  yields  the  lower  positive  tetra-pyramid ,  which  con- 
sists of  the  two  faces  in  front,  and  the  upper  positive  tetra-pyramid 
made  up  of  the  other  two  faces.  In  like  manner  the  negative 
hemi-pyramid  yields  the  upper  and  lower  negative  tetra-pyramids. 
Their  symbols  are  given  in  the  tabulation  on  page  133. 

Hemi-clinodomes.  Obviously,  the  clinodome  is  now  divided 
into  an  upper  and  a  lower  hemi- clinodome. 

Tetra-domes.  The  hemi-orthodomes  yield  upper  and  lower 
tetra-domes. 

Hemi-prisms.  Each  prism  now  yields  two  hemi-prisms,  desig- 
nated as  front  and  rear  forms. 

Pinacoids.  The  clinopinacoids  must  on  account  of  the  sym- 
metry occur  with  the  full  number  of  faces,  namely,  two.  The  basal 
and  orthopinacoids  each  yield  two  forms.  In  the  case  of  the  basal 
pinacoid,  they  are  designated  as  upper  and  lower  forms,  while  the 
terms  front  and  rear  are  employed  for  the  orthopinacoids. 

The  forms  and  their  general  symbols  may  be  tabulated  as 
follows : 

*)     Clinohedral group  of  Dana. 


MONOCLINIC    DOMATIC    CLASS. 


133 


FORMS 

SYMBOLS 

Weiss 

Naumann 

Miller 

Tetra- 
pyramids 

Unit 
and 
Modified 

|       K     :    *    :    ™*}    u        7 

,mP 

\hh~t  \ 
\hhi\ 
\hhl\ 
\ithi\ 

I 

2 

mP 

2     U 

mP 

u 

2 

mP 

2 

-(              2             J 

Clino 

,    (na'  :  b  :  me: 

• 

mPn' 

{hkl} 

\liki\ 
\hki\ 
\hkl\ 

i 

2 

.  mPn' 

\       2      u 
mPn' 

±  1               2                  ">l 

2        U 

mPri 

I                2 

Ortho 

.    (a'  \  nb  \  m£] 

r       mPn 

.   ?nPn 

+  —  u 

mPn 

\khl\ 
\khi\ 
\khi\ 

\khl\ 

±{               2              \11'1 

u 

2 

mP~ii 

~~T~l 

Hemi- 
prisms 

Unit 

r«':Z:«*l 

OOP 

\IIO\ 

\1IO\ 

2     f 
OOP 

1      *      V'1 

T 
2 

Clino 

{na'  :  5:  00^1  , 

f  oo  /V 

\hko\ 
\liko\ 

;    '   f 

CcPn' 

1     ,     K'' 

\           2         ' 

Ortho 

f  a'  :  nb  :  oo^l  f 

(  ccPn 

\kho\ 
\kho\ 

\       2     f 
oo  Pn 

I      *      P 

I      *      ' 

134 


MONOCLINIC    SYSTEM. 


FORMS 

SYMBOLS 

Faces 

Weiss 

Naumann                 Miller 

Hemi-clinodomes 

foo«'  :  b  :  mc\ 

r 
i 

| 

1 

mPte' 

\ohl\ 
\ohl 

!•   2 

j 

7( 
2 

mPac' 

[                2                J  "'  l 

L 

2 

Tetra-orthodomes 

(a':&l:mt\      , 

mP^z 

i               ~, 

\hol\ 

\hol\ 
\hol\ 
\liol\ 

1 

. 

I 

2        U 

mPoo 

1           2       l 

mPtt 

-(     *     r1 

2        ^ 

mP& 

2 

Clinopinacoid 

O0«'  :  b  :  aoc 

Apparently  holohedral 

Pinacoids    - 

Basal 

f  aoa'  :  oob  :  c]        7 

r    OP 

1  ^, 

\OOI\ 

\  ooi  \ 

I  Uy    / 

L      2      ) 

Ortho 

f«':  005:00*] 

ccPac 

\IOO\ 
\IOO\ 

2        S 

oo  Poo 

(                2                \S'> 

T 

2 

Fig.  3.«8. 


OP 
Combination.      Fig.  388.     c        — 21,     {ooi}; 


ooPoo 


a  = 
ocP 


/,     jioo|,   and 


OOPX) 


r,     j  100  J  ;   ;;/    = 


OOP  -       .  Pet' 

/,     {-no},    and  — —    r,    JUO{;    q  =    — —  u, 


{on};     v  =  +   -y-  /, 
Tetrathionate  of  potassium,  KaS4Oe. 


MONOCLINIC     SPHENOIDAL    CLASS. 


135 


3.     MONOCLINIC  SPHENOIDAL    CLASS.1) 

( Hemimorphism. ) 

Symmetry.     A    polar   axis   of   binary 
symmetry  is  the  only  element  of  symmetry  / 

present,  figure  389.  The  forms  of  this 
class  are  hemimorphic  along  the  b  axis, 
which  is  now  a  singular  axis.  XNX  / 

Those  forms  which  extend  parallel  to  Fig.  389. 

b  axis,   the  orthodomes,    basal   and  orthc- 

pinacoids,  are  unaltered.      The  others  yield  right  and  left  forms,  for 
example,  the  unit  prism  yields  the  right  and  left  hemi-prisms. 

The  forms  and  their  symbols  are  tabulated  as  follows: 


SYMBOLS 


\V\ks 

Naumann 

Miller 

j 

••mces 

mP 

\    i 

\  h  h  7( 

- 

mP 

\  /i  11  i  j 

Unit  and 

,          .  f  a'  :  fi  :  me  1 

) 

\       I 

2 

\in  ., 

Modified 

2          \ 

mP 

\  h  h  l\ 

2 

mP 

2 

)    itf  Itf   lr\ 

\hhl\ 

Tetra- 

Clino 

4-  <r    /  1 

mPu' 

2 
2 

\hkl\ 

\hkl\ 

pyramids  " 

I              ^ 

mPri 

{     7       7       7) 

•"    2 

(Sphenoids) 

? 

2 

,  mPri 

\hkl\ 

^777) 

I          /         2 

\hkl\ 

mPn 

+  r  •  

^nPTi 

\khl\ 

5  Jy  J^J  "l 

Ortho 

,  (  naf  \nJ>  :  m£\ 

-h  T    / 

1  '  * 

j>^/?/   | 

I     /  ,  1   \ 
(               2               } 

2 

vi  F  n 

2 

i^y^/S 
\klil\ 

Dan.i  terms  this  class  the  hemimorphic  group. 


136 


MONOCLINIC    SYSTEM. 


FORMS 

SYMBOLS 

] 

Faces 
2 

Weiss 

Naumann 

Miller 

Hemi- 
prisms 

Unit 

f  a'\b\  Cfoc  } 

1*    J 

r     <XP 
If 

\IIO\ 

\ilo\ 

?   2 
\    ,  ™P 

r  ,  t 

2            J 

(       *         2 

Clino 

(na-:S:acc\ 

\        tePn' 
\  r~  — 

oo  Pn' 

\hko\ 
\hko\ 

''l{               2               j 

(!           2 

Ortho 

,  [a'  :  ub  :  ace] 

f        oo  Pn 

\          *¥* 

\kho\ 
\kho\ 

7  aoPn 

r'l(              2               \ 

(    *          2 

Hemi-clinodomes 

,  f  O0#'  :  b  :  mc\ 

j-    / 

f     mPoo  ' 

\ohl\ 
\ohl\ 

"2 

1  mPoo  ' 

r'l{                 *                \ 

r    2 

Clinopinacoids 

(  <Xa':  1  :  Oocl 

r   oopco' 

\OIO\ 

\olo\ 

' 

2 

7  ooPoc' 

/     ,         / 

1                2                J 

r     2 

Hemi  -orthodomes 

4-  a'  :  oo  b  :  me 

\ 
\ 

\  Apparent!) 

i 

r  holohec 

Iral 

Pina- 
coids 

Basal 

aoa':  oob  :  c 

Ortho 

a'  :  cob  :  GCC 

Combinations. 

Figures  390  and  391. 
a  =  OoPoo,  |  looj ;  ;  c  = 
OP,  Jooij;  r=  -Poc, 

OOP 

!  ioi  j;      p   --  :  r 
no  =  I 


2 
OCP 


Fig.  390. 


Fig.  391. 


POO' 

q  ----  I  -—  , 


MONOCLINIC    SPHENOIDAL    CLASS.  137 

Poc' 

{on}  (figure  390);  q  =  r ,  {on|  (figure  391).  These  combi- 
nations occur  on  tartaric  acid,  C4H6O6.  The  above  figures  represent 
left-  and  right-handed  crystals. 


TRICLIN1C  SYSTEM. 


+  c 


Crystallographic   Axes.      This  system  includes  all  forms  which 
can   be  referred   to  three    unequal   axes   intersecting   each   other   at 

oblique  angles.  The  axes  are  designated 
as  in  the  orthorhombic  system,  namely, 
«,  brachydiagonal;  b,  macrodiagonal;  and 
c,  vertical  axis.  From  this  it  follows 
that  one  axis  must  be  held  vertically,  a 
_j  ^/O*'  second  is  directed  toward  the  observer, 

and  then  the  third  is  inclined  from  right 
to  left  or  vice  versa.  Usually  the 
brachydiagonal  is  the  shorter  of  the  two 
lateral  axes.  Figure  392  shows  an  axial 
cross  of  the  triclinic  system.  The  oblique 
angles  between  the  axes  are  indicated  as 
follows:  b  /\  c  =  a,  a  A  c  =  (3,  and  a  A  b 
=  y.  The  elements  of  crystallization 
consist  of  the  axial  ratio  and  the  *hree  angles  a,  /?,  and  y,  page  8. 

Classes  of  Symmetry.     There  are  but  two  classes  of  symmetry 
in  the  triclinic  system,  namely: 

1.  Pinacoidal  class        (Holohedrism). 

2.  Asymmetric  class     (Hemihedrism). 

The  first  is  the  important  class. 


-c 

Fig.  3,)2. 


\ 


V  1-^ 

Fig.  393. 


/.     PINACOIDAL  CLASS.2) 

( Holohedrism. ) 

Symmetry.  A  center  of  symmetry  is  the 
only  element  present.  Hence,  forms  can  con- 
sist of  but  two  faces,  namely,  face  and  parallel 
counter-face.  This  is  represented  diagrammat- 


!)     Also  termed  the  anorthic,  asymmetric  or  clinorhomboidal  system. 
2)     Normal  group  of  Dana. 


[O] 


PINACOIDAL    CLASS. 


139 


ically  by  figure  393,  which  shows  a  triclinic  combination  projected 
upon  the  plane  of  the  a  and  5  axes. 

Tetra-pyramids.  As  already  shown,  triclinic  forms  consist  of 
but  two  faces.  Therefore,  since  the  planes  of  the  crystallographic 
axes  divide  space  into  four  pairs  of  dissimilar  octants,  it  follows  that 
four  types  of  pyramidal  forms  must  result.  These  are  spoken  of  as 
tetra-pyramids  .^  There  are,  hence,  four  tetra-pyramids,  each  cut- 
ting the  axes  at  their  unit  lengths.  The  same  is  also  true  of  the 
modified,  brachy-  and  macro- bipyramids.  That  is  to  say,  the  var- 


-c 


Fig.  394. 


_ 


Fig.  395. 


ious  bipyramids  of  the  orthorhombic  system,  on  account  of  the  obliq- 
uity of  the  three  axes,  now  yield  four  tetra-pyramids  each.  They  are 
designated  as  tipper  right,  upper  left,  lower  rig-Jit,  and  lower  left 
forms,  depending  upon  which  of  the  front  octants  the  form  encloses. 
Naumann  indicates  them  in  turn  by  P',  'P,  P.,  ,P.  The, general 
symbols  for  all  types  are  given  in  the  tabulation  on  page  141.  Figure 
394  shows  the  four  unit  tetra-pyramids  in  combination. 

Henri-prisms.  Obviously  the  prisms  are  now  to  be  designated 
as  rig-ht  and  left  forms.  These  two  forms  are  in  combination  with 
the  basal  pinacoid  in  figure  395. 


In  reality,  tetra-bipyramids,  see  page  127. 


140 


TRICLINIC    SYSTEM. 


Hemi-domes.     All  domes  now  consist  of  but  two  faces.     Hence, 
we  may  speak  of  rig-Jit  and  left  hemi-br achy  domes,  and  upper  and 


Fig.  396. 


Fig.  397. 


lo^ver  hemi-macrodomes .  These  forms  are  shown  in  combination 
with  the  macro-  and  brachypinacoids,  respectively,  in  figures  396  and 
397- 

Pinacoids.     These  forms  occur  with  their  usual  number  of  faces 
and  are  designated,  as  heretofore,  by  the  terms  basal,  brachy-,  and 


I/ 


Fig.  398. 

macropinacoids,  depending  upon  the  fact  whether  they  extend  paral- 
lel to  the  c,  a,  or  b  axis.  Figure  398  shows  these  pinacoids  in 
combination. 


PINACOIDAL    CLASS.  14  L 

The  various  forms  and  symbols  are  given  in  the  following  table: 


FORMS 

SYMBOLS 

Weiss 

Naumann 

Miller 

' 

f       a  :     b  :    c 

P' 

{/£/] 

Unit 

a  :  -b  :    c 
a:     b\-c 

'P 
P, 

1/7/j 

(      a:-b:-e 

<P 

\i7i\ 

f    a  :     b  :    me 

mP' 

\khl\ 

Modified 

a  :  -b  :     me 
a  :     b  \  -me 

m'P 
mP, 

\hhl\ 
\hhl\ 

Tetra- 
pyramids 

• 

a  :  -b  :  -me 

m,P 

\hhl\ 

f    na  :     b       me 

mP'n 

\hkl\ 

Brachy 

na  :  -b       me 
na  :     b     -me 

m'  Pn 
mP,n 

\hkl\ 
\hkl\ 

na  :  -b     -me 

m,Pn 

\hkl\ 

f    a  :     nb       me 

mP'Ti 

\khl\ 

Macro 

\    a  :  -nb       me 
a  :     nb     -me 

m'Pn 
mP,n 

{khl\ 

\khi\ 

s. 

a  :  -nb     -me 

m.Pii 

\khi\ 

Hemi-prisms  - 

Unit 

(      a  :     b  :  OC  c 
\     a  :  -b  :  ooc 

OOP: 

oo  :P 

\IIO\ 
\IIO\ 

Brachy 

{na  \    b  :  oc  c 
na  :  -b  :  ccc 

*™ 

\hko\ 
\hko\ 

Macro 

^ 

!a  :    nb  :  OOc 
a  :  -nb  :  Ooc 

oo:p7i 

\kho\ 
\klio\ 

Hemi-domes  • 

Brachy 

f   ooa  :    b  :  me 
\   ooa  :  -b  :  me 

m.P'o^ 
m  '  P,do 

\ohl( 
\o~hl\ 

Macro 

(   a  :  ocb  :    me 
\    a  :  0/cb  :  -me 

m'P'  oo 

\hol\ 
\hol\ 

Pinacoids 

r  Basal 

oc  a  :  oo  b  :  c 

OP 

\OOI\ 

Brachy 

oca  :  b  :  occ 

ooPoo 

\OIO\ 

Macro 

a  :0c5  :  ccc 

oo  Pod 

\IOO\ 

142 


TKICLINIC    SYSTEM. 


Relationship  of  forms.  The  diagram  on  page  120  may  be  used 
for  this  purpose,  if  we  bear  in  mind  that  bipyramids,  prisms,  and 
domes,  indicated  in  the  same,  now  represent  four  tetra-pyramids, 
two  hemi-prisms,  and  two  hemi-domes,  respectively. 

Combinations.      Figure  399.      x   :  =  P'|ini;    r  =   'P  \ni\\ 


Axinite,  HCa8Al2BSi4O16. 


Fig.  399. 


Fig.  400. 


Fig.  401. 


Figures  400  and  401.  m  =  ooP;|uoj;  M  =  oo'PjuoJ; 
b  =  ooPooioioS;  c  =  OP{ooi};  x  =  ,P,oc{  io~i  |  ;  0  =  P,{ui~}; 
r  ==  iP,  00)403  {  -  Albite,  NaAlSi308. 


2.    ASYMMETRIC  CLASS.1) 


(Hemihedrism.  ) 

Symmetry.  This  class  has  no  element  of 
symmetry.  This  is  clearly  shown  by  figure 
402. 

Forms,  Since  the  center  of  symmetry  is 
lost,  it  follows  that  one  face  may  constitute  a 
form  and,  hence,  this  class  possesses  ogdo- 
pyramtds,  tetra-prisms  ,  and  tetra-domes  .  The  forms  are  desig- 
nated as  in  the  preceding  class,  indicating,  however,  whether  the 
octant  in  which  the  form  lies  is  in  front  or  not.  For  example, 


Fig.  402. 


i )    Dana  calls  this  the  asymmetric  group. 


ASYMMETRIC    CLASS. 


143 


-  a  :  I)  :  c,  P'r,  j  1 1 1  \ ,  is  an  ogdopyramid  occupying 
the  upper  right  rear  octant,  while  a  :  b  :  -c,  P,/~, 
j  i  il } ,  is  another  in  the  lower  right  front  octant. 

The  various  forms  and  their  symbols  are  easily 
deduced  from  those  given  on  page  141. 

Combinations.     Figure    403.     a  ooPoc    /, 

jiooj;  a'  =  ooPoo  r,  |ioo|;  b  =  OcPdo  r,  joio(; 
b'  =  ooPoo  /,  Jolo};  c  =  OP^,  jooi|;  c  =  OP/, 
JooTj,  ?^  =  P'2r,  jl22(;/  =  P^/,  \ioT\;  strontium 
bitartrate,  Sr(C4H4O6H)2.4H2O. 


COMPOUND   CRYSTALS. 


Parallel  Grouping.     Oftentimes  two  crystals  of  the  same  sub- 
stance are  observed  to  have  so  intergrown  that  the  crystallographic 

axes  of  the  one  in- 
dividual are  parallel 
to  those  of  the  other. 
Such  an  arrange- 
ment of  crystals  is 
termed  parallel 
grouping.  Figures 
404  and  405^  show 
such  groups  of  quartz 
and  calcite,  respect- 
Fig.  404.  Fig.  405.  ively.  Occasionally, 

crystals    of   different 
substances  are  observed  grouped  in  this  way. 

Twin  Crystals.  Two  crystals  may  also  intergrow  so  that,  even 
though  parallelism  of  the  crystals  is  wanting,  the  growth  has,  never- 
theless, taken  place  in 
some  definite  manner. 
Such  crystals  are 
spoken  of  as  twin  crys- 
tals, or  in  short, 
twins.  Figure  406 
illustrates  such  a  twin 
crystal  commonly  ob- 
served on  staurolite. 
In  twin  crystals  both 
individuals  have  at  least  one  crystal  plane  or  a  direction  in  common. 
Figure  407  shows  a  twinned  octahedron.  The  plane  common  to 
both  parts  is  termed  the  composition  plane.  In  general,  the 
plane  to  which  the  twin  crystal  is  symmetrical  is  the  twinning 
plane.  In  some  instances,  composition  and  twinning"  planes 
coincide.  Both,  however,  are  parallel  to  some  possible  face  of  the 


Fig.  406. 


Fig.  407. 


After  Tschermak  and  Melczer,  respectively, 


[144] 


COMPOUND    CRYSTALS. 


145 


crystal,  which  is  not  parallel  to  a  plane  of  symmetry.  The  line  or 
direction  perpendicular  to  the  twinning  plane  is  the  twinning-  axis. 
A  twinning  la^v  is  expressed  by  indicating  the  twinning  plane  or 
axis. 

Twin  crystals  are  commonly  divided  into  two  classes:  i)  Con- 
tact or  Juxtaposition  twins,  and  2)  Penetration  twins.  ^  These 
are  illustrated  by  figures  407  and  406,  respectively.  Contact  twins 
consist  of  two  individuals  so  placed  that  if  one  be  rotated  through 
1 80°  about  the  twinning  axis  the  simple  crystal  results.  In  penetra- 
tion twins  two  individuals  have  interpenetrated  one  another.  If  one 
of  the  individuals  be  rotated  through  180°  about  the  twinning  axis 
both  individuals  will  occupy  the  same  position. 

Contact  and  penetration  twins  are  comparatively  common  in  all 
systems.  In  studying  twins,  it  must  be  borne  in  mind,  as  pointed  out 
on  page  3,  that  the  two  individuals  may  not  be  symmetrical  owing 
to  distortion.  Re-entrant  angles  are  commonly  indicative  of  twinning. 

Common  twinning  laws.  Cubic  System.  The  most  common 
law  in  the  cubic  system  is  that  known  as  the  spinel  law,  where  the 
twinning  plane  is  parallel  to  a  face  of  an  octahedron,  O{in}. 
Figure  407  shows  such  a  twin  crystal  of  the  mineral  spinel,  while 
figure  408  shows  a  twinned  octahedron  of  magnetite.  A  penetration 


Fig.  408. 


Fig.  409. 


Fig.  410. 


twin  of  fluorite  is  shown  in  figure  409.  Here,  two  cubes  interpen- 
etrate according  to  the  above  law.  Figure  410  shows  a  penetration 
twin  of  two  tetrahedrons,  observed  on  the  diamond.  The  twinning 
plane  is  a  plane  parallel  to  a  face  of  a  cube,  OoOoo{ioo},  which  in 


i)    Also  designated  as  reflection  and  rotation  twins,  because  they  are  symmetrical  to  a  plane  and 
an  axis,  respectively. 

V?**A**  vV 

OF    THE 

UNIVERSITY 

CF 


146 


COMPOUND    CRYSTALS. 


the  hextetrahedral  class  is  no  longer  a  plane  of  symmetry.  The  axes 
are  parallel  in  the  two  individuals  and  since  the  twinning  tends  to 
give  the  twin  the  symmetry  of  the  hexoctahedral  class,  such  twinned 
crystals  are  sometimes  spoken  of  as  supplementary  twins. 

Of  the  minerals  crystallizing  in  the  dya- 
kisdodecahedral  class,  pyrite  furnishes  excel- 
lent twins.  Figure  411  shows  a  penetration 
twin  of  two  pyritohedrons  of  this  mineral. 
Such  twins  are  often  known  as  crystals  of 
the  iron  cross.  A  plane  parallel  to  a  face  of 
the  rhombic  dodecahedron,  OcO  {nof ,  is  the 
twinning  plane.  Such  a  plane  is  parallel  to 
the  secondary  planes  of  symmetry  which  are 
not  present  in  this  class. 


Fig.  411. 


Hexagonal  System.  Calcite  and  quartz  are  the  only  common 
minerals  belonging  to  this  system  which  furnish  good  examples  of 
twinning.  These  minerals  belong  to  the  ditrigonal  scalenohedral  and 
trigonal  trapezohedral  classes,  respectively. 


Fig.  412. 


Fig.  413. 


Fig.  414. 


Upon  calcite  the  basal  pinacoid,  ORjoooi  J,  is  commonly  a  twin- 
ning plane.  Figure  412  illustrates  this  law.1*  A  plane  parallel  to  a 
face  of  the  negative  rhombohedron,  --^R{oii2},  may  also  be  a 
twinning  plane  as  illustrated  by  figure  413.  These  are  the  most 
common  laws  on  calcite  although  crystals  possessing  a  twinning  plane 
parallel  to  a  face  of  the  rhombohedron,  —  2RJO22IJ,  are  also  to  be 
noted. 


Compare  with  figure  187. 


COMPOUND    CRYSTALS. 


147 


Fig.  415. 


Fig.  416. 


The  so-called  Brazilian  law  is  common  on  twins  of  quartz,  figure 
414.  Here,  a  right  and  left  crystal  have  interpenetrated  so  that  the 
twin  is  now  symmetrical  to  a  plane  parallel  to  a  face  of  the  prism 
of  the  second  order,  OoP2  {1120},  compare  figures  253  and  254. 
Figure  414,  as  is  obvious,  possesses  a  higher  grade  of  symmetry  than 
its  component  individuals,  namely,  that  of  the  ditrigonal  scalenohe- 
dral  class. 

Tetragonal  System.  Most 
of  the  twin  crystals  of  this 
system  are  to  be  observed  on 
substances  crystallizing  in  the 
ditetragonal  bipyramidal  class. 
In  this  class  a  plane  parallel  to 
a  face  of  the  unit  bipyramid  of 
the  second  order,  Poo  jouj, 
commonly  acts  as  the  twinning 
plane.  Figures  415  and  416 
show  crystals  of  cassiterite  and  zircon,  respectively,  twinned  accord- 
ing to  this  law. 

Orthorhombie  System.  The  most  common  twins  of  this  system 
belong  to  the  bipyramidal  (holohedral)  class  in  which  any  face  aside 
from  the  pinacoids  may  act 
as  twinning  plane.  Figure 
406,  page  144,  shows  a  pen- 
etration twin  of  staurolite, 
where  the  brachydome,  -fPoc 
{032},  acts  as  the  twinning 
plane.  Figure  417  shows 
the  same  mineral  with  the 
pyramid,  -fPf  5232;,  as  the 
twinning  plane.  Figure  418 

represents  a  contact  twin  of  aragonite.      Here  the  unit  prism,  OoP 
1 1 10},  is  the  twinning  plane. 

Monoclinic  System.  In  this  system,  gypsum  and  orthoclase 
furnish  some  of  the  best  examples.  Figure  419  shows  a  contact 
twin  of  gypsum  in  which  the  orthopinacoid,  ooPoojioo},  is  the 
twinning  plane.  A  penetration  twin  of  orthoclase  is  shown  in  figure 


Fig.  417. 


Fig.  418. 


148 


COMPOUND    CRYSTALS. 


Fig.  419.        Fig.  420. 


420.  Here,  the  orthopinacoid  also  acts  as 
twinning  plane.  This  is  known  as  the 
Karlsbad  law  on  orthoclase. 

Triclinic  System.     Since  there  are  no 
planes  of    symmetry    in    this    system,    any 
plane  may  act  as  the  twinning  plane.     The 
mineral  albite  furnishes  good  examples.     In 
figure  421,  the  clinopinacoid,   OoPoo  |oioj, 
acts  as  the  twinning  plane.       This    is  the 
albite  lazv.    Another 
common  law  is  shown 
by  figure  422.   Here, 
the    basal    pinacoids 
of     both    individuals 
are  parallel,  thecrys- 
tallographic    b     axis 

acting  as  the  twinning  axis.     This  is  known  as  the 
fiericline  law. 

Repeated  twinning.  In  the  foregoing,  crystals  consisting  of  but 
two  individuals  have  been  discussed.  Intergrowths  of  three,  four, 
five,  etc.,  individuals  are  termed  threelings,  fourlings,  fivelings,  and  so 


Fig.  421. 


Fig.  423. 


Fig.  424. 


Fig.  425. 


on.  Poly  synthetic  and  cyclic  twins  are  the  result  of  repeated  twin- 
ning. In  the  polysynthetic  twins  the  twinning  planes  between  any  two 
individuals  are  parallel.  This  is  illustrated  by  figures  423  and  424 
showing  polysynthetic  twins  of  albite  and  aragonite,  respectively. 1J 
If  the  individuals  are  very  thin  the  re-entrant  angles  are  usually 


1 )     Compare  with  figures  418  and  421. 


COMPOUND    CRYSTALS. 


149 


indicated  by  striations.  The  cyclic  twins  result  when  the  twinning 
planes  do  not  remain  parallel,  as  for  example  when  adjacent  faces  of 
a  form  act  as  twinning  planes.  This  is  shown  by  the  cyclic  twins  of 
rutile,  figure  425,  in  which  adjoining  faces  of  the  unit  bipyramid  of 
the  second  order  OoP2  joi  i  \  act  as  twinning  planes. 

Mimicry.      As    a 

result  of  repeated  twin- 
ning, forms  of  an  appar- 
ently higher  grade  of 
symmetry  usually  re- 
sult. This  is  especially 
common  with  those 
substances  possessing 
pseudosymmetry,  page  Fig.  426. 

94.     Figure  426  shows 

a  trilling  of  the  orthorhombic  mineral  aragonite,  CaCO3,  which  is 
now  apparently  hexagonal  in  its  outline.  In  figure  427  the  cross- 
section  is  shown.  This  phenomenon  is  called  mimicry. 


-t  >' 


oi"7 


TABULAR  CLASSIFICATION 
SHOWING  THB  ELEMENTS  OF  SYMMETRY 

AND  THE  SIMPLE  FORMS 
OF  THB  THtRTY-TWO  CLASSES  OF  CRYSTALS. 


CUBIC  SYSTEM. 

Classes  of  Crystals,  I  to  5. 

(Page  150.) 


TABULAR  CLASSIFICATION  SHOWING   THE  ELEMENTS  OF  SYMMETRY  A* 


/.     CUBIC 


SYMMETRY 

Planes 

Axes 

ci  \  ci  '.  a 

ft  *  n  '  QC  n 

CLASS 

_* 

>, 

IM 

_a 

rt 

T3 

a 

a> 

0 

ocO 

~ 

^H 

w^ 

u 

a, 

1 

i/7/i 

j//o|- 

\W-  "  I     Vw    ^    t 

r 

> 

t- 

i.   Jttexoetahedral 

(  Holohedrisni] 

3 

6 

3 

4 

6 

I 

Octahedron 

Ht 

Dodecahedron 

Tetrahe- 

2.    Hextetrahedral 

__ 

6   - 

4 

3 

_ 

drons 

(  Tetrahedral  Hemihedrisni} 

(Polar, 

3.    Dyakisdodecahedral 

4 

3 

I 

(  Pyritohedral^  Hemihedrism  ) 

4.     Pentagonal 

Icositetrahedral 

— 

— 

3 

4 

6 



(  Plagihedral  Hemihedrism  ) 

5.     Tetrahedral 

Tetrahe- 

Pentagonal 
Dodecahedral 

— 

— 

— 

4 

3 

.  

drons 

(Polar) 

(  —  T~  ) 

(  Tetartohedrisni} 

1)     The  blank  spaces  indicate  that  the  forms  are  apparently  holohedral. 


[1501 


9  THE  SIMPLE  FORMS  OF   THE   THIRTY-TWO   CLASSES  OF  CRYSTALS. 

SYSTEMS 

j 

FORMS 

REPRESENT- 
ATIVE 

DOrt  :  OOrt 
TOOOC 

100  \ 

a  '  :  a  :  ma 
mO 
\hhl\ 

a  :  ma  :  ma 
mOm 

\hu\ 

a  :  ma  :  ooa 
acOm 
\hko\ 

a  :  na  :  ma 
mOn 

\hki\ 

Cu** 

xahedron 

I 

/ 

^Tetragonal 
Trisoctahe- 
dron 

\\ 
Tetrahexa- 

hedron 

!Hexoctahe- 
dron 

Galena 
(PbS) 

Trisoctahe- 
dron 

i 

I 

Tetragonal 
Tristetrahe- 
drons 

(±) 

Trigonal 
Tristetrahe- 
drons 

(±) 

Hextetrahe- 
drons 

(±) 

Tetrahedrite 
(Cu2,Fe,Zn)4 
(AS,Sb)2S7 

i 

Pyritohe- 
drons 

(±) 

Dyakis- 
dodecahedrons 

(±) 

Pyrite 
(FeS2) 

Pentagonal 
Icositetrahe- 
drons 
(r,D 

Sal 
Ammoniac 
(NH4C1) 

Tetragonal 
Tristetrahe- 
drons 

(±) 

Trigonal 
Tristetrahe- 
drons 

(±) 

Pyritohe- 
drons 

(±) 

Tetrahedral 
Pentagonal 
Dodecahedrons 

±r,    ±1 

Sodium 
Chlorate 
(NaClO3) 

[150] 


HEXAGONAL  SYSTEM. 

Classes  of  Crystals,  6  to  II. 

(Page  151.) 


151 


2.     HEXAQO 


CLASS 

SYMMETRY 

- 

c 

• 

Planes 

Axes 

A 

a  :  oca:  a:mc 
mP     yr 

j/O/7    { 

2«  :  2r/  :  (t  :  me 
mP2 
\hh2hl\ 

\na 

Principal 

Secondary 

Intermediatt 

* 

6.     Dihexagonal 

Hexagonal 

Hexagonal    ! 

Bipyramidal 

I 

3 

3 

I 

- 

3  +  3 

i 

Bipyramids 

Bipyramids 

R 

|     (  Holohedrism  ) 

First  order 

Second  order 

7.     Dihexagonal 
Pyramidal 

3 

^? 

Hexagonal 
Pyramids 

Hexagonal 
Pyramids 

Di 

i: 

f  Holohedrism   with  1 

(Polar) 

First  order 

Second  order 

(    Hemimorphism   j 

(»,  fl 

(*,  /) 

8.    Ditrigonal 
Bipyramidal 

I 

i 

Trigonal 
Bipyramids 

E 

Bi 

f       Trigonal      1 

'    (Pc/ar) 

First  order 

/•  Tj)  Q  ' 

(  ±  ) 

9.    Ditrigonal 

Rhombohe- 

Sc 

Scalenohedral 

? 

1 

3       J 

drons 

f  Rhombohedral  1 

First  order 

rt  L  HemjJtedr^sm  } 

(±) 

) 

10.     Hexagonal 

Bipyramidal 

I 

I 

_ 

i 

Bi 

f     Pyramidal     1 

ri 

/    ^  Hemihedrism  j 

n.  Hexagonal 

H 

Trapezohedral 

_ 

_    — 

I 

3  +  3    ' 

Ti 

(Trapezohedral  1 

Hemihedrism  j 

5" 

-  - 

II.  SYSTEM.                                                                                  i£ 

FORMS 

REPRESENTATIVl 

a,  \a\nu 

ocP 

ocP? 

na  :  pa  :a  :  cce  oca  :jCa:Oca  :c 
ooP?^      9           OP 

kl\ 

\IOIO\  '~ 

\II20\ 

\hiko\ 

\OOOI  \ 

agonal 
ramids 

Hexagonal 
Prism 
First  order 

Hexagonal 
Prism 
Second  order 

Dihexagonal 
Prisms 

i 
Basal 
Pinacoid 

Beryl 

(Be3Al2(Si08)fl 

:agonal 
trnids 

Basal 
Pinacoids 

O,  /) 

Zincite 
(ZnO) 

gonal 
•amids 

h) 

Trigonal 
Prisms 
First  order 

Ditrigonal 
Prisms 



nohe- 
ons 

t) 

Calcite 
(CaCO3) 

gonal 
•amids 
order 

1) 

Hexagonal 
Prisms 
Third  order 

Apatite 
(Ca5Cl(P04)3) 

Barium  Stibio 

igonal 
;zohe- 

- 

tartrate       a  n  < 
Potassium    Ni 
trate 

/) 

f  Ba  (SbO)2  ' 

KN03      J 

HEXAGONAL  SYSTEM. 

(Continued) 
Classes  of  Crystals,  12  to  17. 

(Page  152.) 


152 


2.     HEXAGC 

(Co, 


CLASS 


12.  Ditrigonal 
Pyramidal 

Trigonal  Hemi- 
hedrisnt  wifli 
Hemimorphism 

1V"T 

13.  Hexagonal 
Pyramidal 

f  Pyramidal  Heini-  1 
//  c  drism      with  i 


A-  [^  Hemimorph 


14.     Trigonal 
Bi  pyramidal 

r  Trigonal  "I 

L  Tetartohedrism  } 

1  5.  Trigonal 
Trapezohedral 

r  Trabezohedral  1 
r7  \Jfcta 


1 6.  Trigonal 
Khombohedral 

f  Rhombohedral  \ 
L,  jfTetartohedrism  j 

17.  Trigonal 
Pyramidal 

( Ogdohedrism ) 


SYMMETRY 

Planes                           Axes 

Center 

a:  do  a:  a:  me 
mP 

\IOII\ 

2a  :  2a  :  a  :  Die  / 
mP2 
\  h  h~2~hl\ 

Piincipal 

X 

rt 
•o 
c 
o 

Intermediate 

• 

A 

^P 

- 

i 

(Polar) 

- 

Trigonal 
Pyramids 
First  order 

Hexagonal 
Pyramids 
Second  order 

I 

i 

(Polar.) 

- 

- 

Hexagonal 
Pyramids 
First  order 

Hexagonal 
Pyramids 
Second  order 

(".  0 

I 

- 

Trigonal 
Bipyramids 
First  order 

Trigonal 
Bipyramids 
Second  order 

— 

- 

i          3 

(  Polar) 

- 

Rhombohe- 
drons 
First  order 

Trigonal 
Bipyramids 
Second  order 

- 

- 

I. 

j         ,  •> 

1 

I 

Rhombohe- 
drons 
First  order 

Rhombohe- 
drons 
Second  order 

- 

- 

- 

I 

(Polar) 

- 

Trigonal 
Pyramids 
First  order 
(+«,  ±1) 

Trigonal 
Pyramids 
Second  order 

i 

: 


\L  SYSTEM.                                                                                   152 

'ted} 

FORMS 

a  :  a  :  me  a  :  oca  :  a  :  oc  r 

2a  :  2  a  :a:Occ 

na  \pa  :  a  :  ccc 

OC«:(X«:00«:6- 

REPRESENTATIVE 

iPn 

OOP 

OOP2 

acPn 

OP 

ikl\ 

\IOIO\ 

\II20\ 

\hiko\ 

•  0001  \ 

T  o  ur  mal  in  e 

-igonal 
amids 

Trigonal 
Prisms 

Ditrigonal 
Prisms 

Basal 
Pinacoids 

A13B205H2 

fAl  Mg  Fe  ] 

v,  +/) 

First  order 

(«,  /) 

3  '    2    '    2   > 

Li,  Na,   HJ9 

(SiOj4 

agonal 
amids 
d  order 

- 

Hexagonal 
Prisms 
Third  order 

Basal 
Pinacoid 
//,  / 

Strontium  Anti- 
monyl  -  tartrate 
Sr(SbO)2(C4H4 

gonal 

Trigonal 

Trigonal 

Trigonal 

ramids 

Prisms 

Prisms             Prisms 

d  order 

First  order 

Second  order  Third  order 

igonal 
>ezohe- 
rons 

Trigonal 
Prisms 
Second  order 

Ditrigonal 
Prisms 

Quartz 

(Si02) 

nbohe- 

Hexagonal 

rons 

Prisms 

Dioptase 

d  order 

Third  order 

(H2CuSiO4) 

',  ±1 

(±) 

gonal 

•amids 

d  order 
r,  u; 
r,  I; 

Trigonal 
Prisms 
First  order 

Trigonal 
Prisms 
Second  order 

Trigonal 
Prisms 
Third  order 

/I                  1      7\ 

Basal 
Pinacoids 

Sodium 
Periodate 
(NaI04-f3H20) 

/,  u; 

• 

— 

(  ~r  r,  ~ri) 

'' 

TETRAGONAL  SYSTEM. 

Classes  of  Crystals,  18  to  24. 

(Page  153.) 


153 


3.      TETRA 


CLASS 

SYMMETRY 

Planes 

'Axes 

u 

1 

I 

a  :  a  :  me 
?;/P 
\hhl\ 

a  :  oca  :  > 
;;/Poo 
\hol\ 

1 
o 

e 

£ 

I 

N)  Secondary 

Intermediate 

• 

• 

18.     Ditetragonal 
Bi  pyramidal 

(  Holohedrism  )             . 

2 

I 

2  +  2 

Tetragonal 
Bipyramids 
First  order 

Tetragoi 
Bipyram 
Second  01 

19.     Ditetragonal 
Pyramidal 

{  Holohedrism  with"\ 
[    Hemimorphism  j 

_5* 

2 

2 

I 
(Polar) 



— 

Tetragonal 
Pyramids 
First  order 

(",  i) 

Tetragoi 
Pyrami( 
Second  or 

(",  i) 

20.     Tetragonal 
Soalenohedrai 

f    Sphenoidal  1 
i  Hemihedrism  i          _*. 

5 

— 

2 



I  +2 

Tetragonal 
Bisphenoids 
First  order 

(+) 

21.     Tetragonal 
15  i  pyramidal 

7i  |'     Pyramidal    1 
i  Hemihedrism  \        ^ 

— 

— 

I 



I 

22.     Tetragonal 
Trapezohedral 

f  Trapezohedral} 
L  Hemihedrism  j        / 

—  ;  — 

I 

2  +  2 

23.    Tetragonal 
Pyramidal 

r  Pyra  midal  Hem  ih  edrism  1 
(^    zt'zV//  Hemitnorphism     u 

— 



I 
(  Polar) 



— 

Tetragonal 
Pyramids 
First  order 

(«,  i) 

Tetragoi 
Pyrainic 
Second  or 

(".  /) 

24.     Tetragonal 
Bisphenoidal 

(  Tetartohedrism) 

7 

— 



— 



I 

— 

Tetragonal 
Bisphenoids 
First  order 

(+) 

Tetragoi 
Bispheno 
Second  or 

(±) 

MAL    SYSTEM. 


153 


FORMS 

7  :  na  :  me 
niPit 

\hki\ 

a  \a  :  ooc 
ocP 
iiio| 

a  :  oc  a  :  00  c  \ 
ocPoo 
|ioo| 

a  :  na  :  GO  c 
00  Pfl 

\hko\ 

oo  a  :  00  a  :  c 
OP 

\00l\ 

REPRESENTATIVE 

)itetragonal 
Bipyramids 

Tetragonal 
Prism 
First  order 

Tetragonal   i 
Prism 
Second  order 

Ditetra- 
gonal 
Prisms 

Basal 
Pinacoid 

Cassiterite 
(Sn02) 

)itetragonal 
Pyramids 

(«,  0 

Basal 
Pinacoids 

(«,  0 

Silver  Fluoride 
(AgF  +  H20) 

Tetragonal 
Scalenohe- 
drons 

(±) 

Chalcopyrite 
(CuFeS2) 

Tetragonal 
Bipyramids 
Third  order 

(+) 

Tetragonal 
Prisms 
Third  order 

(±) 

Scheelite 
(CaWOJ 

Tetragonal 
Trapezohe- 
drons 
(r,  /) 

Nickel  Sulphate 
(NiSO,  +  6H2O) 

Tetragonal 
Pyramids 
Third  order 

±  w>  ±  l 

Tetragonal 
Prisms 
Third  order 

(±) 

Basal 
Pinacoids 
(w,  /) 

Wulfenite 
(PbMoOj 

Tetragonal 
Bisphenoids 
Third  order 

±r,  ±1 

Tetragonal 
Prisms 
Third  order 

(±) 



ORTHORHOMBIC  SYSTEM. 

Classes  of  Crystals,  25  to  27. 

(Page  154.) 


154 


4.     ORTHOR 


CLASS 

SYMMETRY 

i 

S3 

E 

A   L 
1  ^ 

Center 

na  :  b  :  me 
mPn 
\hkl\ 

nd  :  b  :  occ 
00  Pn 
\hko\ 

o 

25.     Orthorhombic 
Bipyramidal 

(Holohedrism} 

I  +  I  +  I 

l  +  I  +1 

I 

Orthorhombic 
Bipyramids 

Orthorhombic 
Prisms 

B 

26.    Orthorhombic 
Pyramidal            r  ,  t 

f        Holohedrism         1 
[  with  Hemimorphism  j 

I 

(P.tlar) 

Orthorhombic 
Pyramids 

(«,  i) 

B 

27.     Orthorhombic 
Bisphenoidal 

(Hemihedrism] 

— 

I  +  I+I 



Orthorhombic 
Bisphenoids 
(r,  I) 

WIC   SYSTEM. 


154 


FORMS 

_ 

:  me 

X 

l\ 

a  :  oc5  ':  me 
mPoo 
\hol\ 

OO<2  :  &  :  ODC 
OcPob 
JOIOJ 

«  :  oc^  :  ace 
OoPoo 
i  iooj 

oca  :  cob  :  c 
OP 
JooiJ 

REPRESENTATIVE 

domes 

Macrodomes 

Brachy- 
pinacoid 

Macro- 
pinacoid 

Basal 
Pinacoid 

Barite 
(BaSOJ 

lomes 
/) 

Macrodomes 
(*,/) 

Basal 
Pinacoids 

(«.  /) 

Calamine 
(Zn2(OH)2SiO,) 

Epsomite 
(MgSOt+7HaO) 

MONOCLINIC  SYSTEM. 

Classes  of  Crystals,  28  to  30. 

(Page  155.) 


155 


5.     MONOC 


CLASS 

SYMMETRY 

0) 

c 

ri 
Ou 

M 

V 

c5 
i 

•na'  :  b  :  me 
mPn' 
\hkl\ 

uu'  :  b  :  ocr 
OcP;/' 
\hko\ 

Oca'  :  b  :  me 
mPoo' 
\ohl\ 

28.     Monoclinic 
Prismatic 

(Ho/ohedrism] 

I 
I 

I 

Hemi-pyramids 

(±) 

Prisms 

Clinodomes 

29.     Monoclinic 
Domatic 

(Hetnihedrism] 



— 

Tetra-pyramids 

(±«.  ±0 

Hemi-prisms 
(J\r) 

Hemi- 
clinodomes 

(«,  i) 

30.     Monoclinic 
Sphenoidal 

(  Hemimorphism  ) 

— 

I 

(/War) 

— 

Tetra-pyramids 

(  Sphenoids) 

(±n  ±0 

Hemi-prisms 
(r,  I) 

Hemi- 
clinodomes 

(r,  /) 

'1C  SYSTEM. 


155 


FORMS 

REPRESENTATIVE 

:  ocb  :  me 
wPoo 

SAO/! 

oc«'  :  ^  :  ocr 
00  POO' 

jo/oj 

a':  oob  :  ccc 
ooPoo 

\IOO\ 

oca'  :  oo^  :  £ 
OP 

{oo/j 

Hemi- 
rthodomes 

(±) 

Clinopinacoid 

Orthopinacoid 

Basal 
Pinacoid 

Gypsum 
(CaS04  +  2H20) 

Tetra- 
rthodomes 

+  w,  +  /) 

Orthopinacoids 
(f,r) 

Basal 
Pinacoids 
(«,  /) 

Tetrathionate  of 
Potassium 

(KAO.) 

Clinopinacoids 
(r,/) 

Tartaric  Acid 
(CtH6Os) 

TRICUNIC  SYSTEM. 

Classes  of  Crystals,  31  and  32. 

(Page  1 56.) 


156 


6.     TRIG* 


CLASS 

SYMMETRY 

1 
^e 

(A 
< 

3 
I 

na  \  b  \  me 
mPn 
\hkl\ 

wa  \  b  :C6c 

mPao' 
\ohl\ 

U 
31.     Triclinic 
Pinacoidal 

(  Holohedrisni) 

Tetra-pyramids 

Hemi-prisms 
(r,/) 

Hemi- 
brachydomes 

C, 

Ogc 

io-pyram 
?^ 

ids 

32.     Triclinic 
Asymmetric 

(  Hemihedrism  ) 







?^ 
n 

Tetra-prisms 
f  r  ,  r     1 
[7/'7r] 

Tetra- 
brachydomes 

r  ?/    u    I 

u 

w                             -< 

7  SYSTEM. 


156 


^ORMS 

»£  :  mt 

OOtf  :  &  :  oc^ 

a  :  006  :  ooc 

a:  005:00^ 

REPRESENTATIVE 

zPoc 

OoPdc 

oo  Poo 

OP 

H6l\ 

So/oj 

j/oo( 

\OOI\ 

lemi- 
rodomes 

Brachy- 
pinacoid 

Macro- 
pinacoid 

Basal 
Pinacoid 

Albite 
(NaAlSi3O8) 

retra- 

rodomes 

Brachy- 

Macro- 

Basal 

Strontium  bitartrate 

pinacoids 

pinacoids 

Pinacoids 

fSr(C4Ht06H)2  +  ] 

/•M 

d,r) 

I          4HS0 

INDEX. 


Acid,  tartaric,  137,   155 
Albite,  8,  142,  148,  156 
Albite  Law,  148 
Alum,  2 
Amorphous  structure,  1 

substances,  2 
Analcite,   24 
Anatase,  99 
•     Anglesite,  8 

Angular  position  of  faces,  14 

Anorthic  system,  138 

Apatite,  68,  151 

Apparently   holohedral    forms,    29 

Aragonite,  8,  120,  147,  148,  149 

Argentite,  24,  25 

Asymmetric  class,   142,  156 

group,  142 

system,  138 

Axes,  crystallographic,  4,  17,  43,  92,  115, 
126,  138 

of  symmetry,  13 
Axial  cross,  4 
Axial  ratio,  7 
Axinite,  142 
Axis,  polar,  16 

singular,  16 

Barite,  154 

Barium  nitrate,  42 

Barium     stibiotartrate     and     potassium 

nitrate,   70,   151 
Beryl,  53,   151 
Bisphenoids,  orthorhombic,  124 

tetragonal,  first  order,  102,   111 
second  order,  111 
third  order,  113 
Boracite,  32 
Brachy  bipyramids,  117 

diagonal,  115,  138 

domes,  118 

prisms, -117 
Brazilian  law,  147 
Brookite,  120 


Calamine,  123,  154 

Calcite,  64,  65,  144,  146,  151 

Cassiterite,  99,  147,  153 

Celestite,  121 

Center  of  symmetry,  14 

Chalcocite,  120 

Chalcopyrite,  105,  153 

Chemical  crystallography,   2 

Chrysoberyl,  120 

Cinnabar,  85 

Classes  of  crystals,  14,  150 

Classification    of   the   thirty-two   classes 

of  crystals,  150 
Clino-axis,  126 

domes,  128 

hemi-pyramid,  128 

prism,   128 

rhombic  system,  126 

rhomboidal   system,   138 
Clinohedral  group,  132 
Closed  forms,  6 
Cobaltite,  36 

Coefficients,  rationality  of,  9 
Combinations,  7 

cubic,  23,  31,  35,  38,  42 

hexagonal,  52,  55,  64,  68,  74,  76,  85, 
89,  91 

tetragonal,  98,  101,  104,  108,  111 

orthorhombic,  120,  123,  125 

monoclinic,  131,  134,  136 

triclinic,  142,  143 
Common  twinning  laws,  145 
Composition  plane,  144 
Compound  crystals,  144 
Congruent  forms,  16 
Constancy  of  interfa^ial  angles,   2 
Contact  twins,  145 
Copper,  24 

Correlated  forms,  15 
Corundum,  64,  65 
Crystal  faces,  2 

forms,  6 

habit,  4 

[157] 


158 


INDEX. 


Crystal  system,  14 

systems,    5 

Crystalline  structure,  1 
Crystallization,    elements    of,    8 
Crystallized  substances,  2 
Crystallographic  axes,  4 

cubic,  17 

hexagonal,  43 

tetragonal,   92 

orthorhombic,   115 

monoclinic,   126 

triclinic,  138 
Crystallography,   2 
Crystals,   1,   2, 

distorted,  3 

formation  of,  1,  2 
Cube,  19 

Cubic  system,  5,  17,  150 
Cyclic  twins,  148 

Dana's  symbols,  11 
Deltoid,  27 
Diamond,  145 
Didodecahedron,  34 
Dihexagonal  bipyramid,  48 

bipyramidal  class,  44,  151 

prism,  50 

pyramidal  class,  53,  151 

pyramids,  54 
Diopside,   131 
Dioptase,  89,  152 
Diploid,  34 
Distortion,   3 
Ditetragonal  bipyramid,  95 

bipyramidal  class,  93,  153 

prism,  96 

pyramidal  class,  99,  153 
Ditrigonal   bipyramidal   class,    56,    151 

bipyramids,  57 

prisms,  59,  72 

pyramids,  71 

pyramidal  class,  70,   152 

pyramidal  tetartohedrism,  70 

scalenohedral    class,    60,    151 
Dodecahedron,   19 

deltoid,   27 

pentagonal,  33 

rhombic,  19 

tetrahedral  pentagonal,  39 


Dolomite,  89 
Domatic  class,  132,  155 
Dyakisdodecahedral  class,  32,  150 
Dyakisdodecahedron,  34 

Elements  of  crystallization,  8 

symmetry,  12 
Epsomite,  125,   154 
Etch-figures,  29 

Faces,  angular  position  of,  14 
Fivelings,  148 
Fluorite,  24,  145 
Form,  crystal,  6 
Formation  of  crystals,   1,  2 
Forms,   congruent,    16 

closed,  6 

correlated,  15 

Enantiomorphous,  16 

fundamental,  6 

hemi-morphic,    16 

open,  6 

unit,  7 

Fourlings,   148 
Fundamental  forms,  6 

Galena,  24,  150 

Garnet,  25 

Geometrical  crystallography,  2 

Gypsum,  8,  131,  147,  155 

Gyroidal  hemihedrism,  36 

Gyroids,   37 

Habit,  crystal,  4 

prismatic,  4 

tabular,  4 
Halite,  24,  29 
Hematite,  64,  65 
Hemi-bipyramid,  127 

clinodomes,  132 

domes,  140 
Hemihedrism,  15 

cubic,  26,  32,  36 

hexagonal,  55 

tetragonal,  101 

orthorhombic,  123 

monoclinic,   132 

triclinic,  142 


INDEX. 


159 


Hemimorphic  forms,  16 
group,  hexagonal,  53 
monoclinic,  135 
orthorhombic,  121 
tetragonal,   99 
Hemimorphism,  16 

hexagonal,  53,  70,  74,  89 
monoclinic,  132 
orthorhombic,  121 
tetragonal,  99,  110 
Hemimorphite>  103 
Hemi-orthodomes,    129 
Hemiprisms,  monoclinic,  132 

triclinic,  139 

Hemiprismatic   system,   126 
Hemipyramids,  127 
Hexagonal  basal  pinacoid,  51 
bipyramidal  class,  65,  151 
bipyramid,  first  order,  45 
second  order,  46 
third  order,   65 
hemihedrisms,  55 
prism,  first  order,  50 
second  order,  50,  72 
third  order,  67,  87 
pyramid,  first  order,  54 
second  order,  54,  71 
third  order,  75 
pyramidal  class,  74,  152 
system,  5,  43,  151,  152 
tetartohedrisms,  76 
trapezohedral  class,  68,  151 
trapezohedrons,    68 
Hexahedron,  19 
Hexoctahedral  class,  17,  150 
Hexoctahedron,  21 
Hextetrahedral  class,  26,  150 
Hextetrahedron,   28 
Holohedral,  apparently,  29 
Holohedrism,  15 

Icositetrahedron,  20 
Incline-face  hemihedrism,  27 
Indices,   Miller's,   12 
lodyrite,  55 
Iron  cross,  146 
Isometric  system,  17 

Juxtaposition   twins,   145 


Karlsbad  law,  148 

Leucitohedron,  20 
Limiting  forms,  23 

Macrobipyramids,  117 

diagonal,   115,   138 

domes,   118 

prisms,  117 
Magnetite,  25,  145 
Miller's  indices,  12 

system,  12 
Mimicry,    149 
Modified  hemi-pyramid,  128 

pyramid,  7 
Monoclinic  domatic  class,  132,  155 

domes,  128 

prismatic  class,  126,  155 

prisms,  128 

sphenoidal  class,  135,  155 

system,  5,  126,  155 
Monoclinohedral   system,   126 
Monosymmetric  system,  126 

Naumann  symbols,  11,  62 
Nepheline,  76 
Nickel  sulphate,  109,  153 
Normal  group,  cubic,  17 

hexagonal,   44 

monoclinic,  126 

orthorhombic,   115 

tetragonal,  93 

triclinic,   138 

Oblique  system,  126 

Octahedron,  19 

Octants,  17 

Ogdohedrism,  15,  89 

Ogdo-pyramids,   142 

Open  forms,  6 

Ortho-axis,  126 

Orthoclase,  131,  147,  148 

Orthohemi-pyramid,  128 

Ortho-prism,  128 

Orthorhombic  bipyramidal  class,  115,  154 

bipyramids,  116 

bisphenoidal  class,  123,  154 

bisphenoids,   124 

domes,  118 


160 


INDEX. 


Orthorhombic  pinacoids,   119 
prisms,  117 

pyramidal  class,  121,  154 
system,  5,  115,  154 

Parallel-face  hemihedrism,  32 

grouping,  144 
Parameters,  5 
Parametral  ratio,  5 
Penetration  twins,   145 
Penta-erythrite,  101 
Pentagonal  dodecahedron,  33 

hemihedrism,  32 

icositetrahedron,   37 

icositetahedral  class,  36,  150 
Pericline  law,  148 
Physical  crystallography,  2 
Pinacoidal  class,  138,  156 
Plagiohedral   group,   36 

hemihedrism,  36 
Planes  of  symmetry,  12 
Polar  axes,  16 
Pole,  53 

Polysynthetic  twins,   148 
Potassium  tetrathionate,  134,  155 
Praseodymium  sulphate,  132 
Prismatic  habit,  4 

system,  115 

Pseudosymmetry,  94,  149 
Pyramid  cube,  21 
Pyramidal  group,  hexagonal,  65 
tetragonal,  105 

hemihedrism,  hexagonal,  55,  65 
tetragonal,  101,  105 

system,  92 

Pyrite,  30,  33,  35,  36,  146,  150 
Pyritohedral  group,  32 

hemihedrism,  32 
Pyritohedron,    33 

Quadratic  system,  91 

Quartz,  8,  85,  144,  146,  147,  152 

Ratio,  axial,  7 

parametral,  5 

Rationality  of  coefficients,  9 
Re-entrant  angles,  148 
Reflection  twins,  145 


Regular  system,  17 
Repeated  twinning,  148 
Rhombic  dodecahedron,  19 
Rhombic  system,  115 
Rhombohedral  group,  60 

hemimorphic,   70 

hemihedrism,  55,  60 

hemimorphic  class,  70 

tetartohedrism,  76,  85 
Rhombohedron  of  the  middle  edges,  62 

— like  forms,  71 
Rhombohedrons,  first  order,  60,  81,  85 

second  order,  86 

third  order,  86 
Rotation  twins,  145 
Rutile,  149 

Salammoniac,   38,    150 
Scalenohedron — like  forms,  71 
Scalenohedrons,  hexagonal,  61 

tetragonal,  103 
Scheelite,  108,  153 
Siderite,  64 

Silver  fluoride,  101,  153 
Singular  axis,  16 
Sodium  chlorate,  42,  150 

periodate,  91,  152 
Sphalerite,  30,  31,  32 
Sphenoidal   group,    orthorhombic,   123 
tetragonal,  102 

hemihedrism,  101,  102 
Spinel,  24,  25,  145 

law,  145 

Staurolite,  144,  147 
Steno,  Nicolas,  4 
Striations,   31,   149 
Strontium   antimonyl-tartarte,   76,   152 

bitartrate,  143,  156 
Structure,  amorphous,  1 

crystalline,  1 
Struvite,  123 
Strychnine  sulphate,  109 
Sulphur,  7,  120 
Supplementary  twins,  146 
Sylvite,  29 
Symbols,  10 

Weiss,  11 
Symmetry,  axes  of,  13,  14 


INDEX. 


161 


Symmetry,  center,  14 
classes  of,  14 
elements  of,  12 
planes  of,   12 

Tabular  habit,  4 
Tartaric  acid,  137,  155 
Tesseral  system,  17 
Tessular    system,   17 
Tetartohedral  group,  cubic,  38 

tetragonal,  111 
Tetartohedrism,  15 
cubic,  38 
hexagonal,  76 
tetragonal,  111 
Tetra-bipyramids,  139 

domes,  monoclinic,  132 
triclinic,  142 

pyramids,  monoclinic,  132 
triclinic,  139 

Tetrathionate  of  potassium,  134,  155 
Tetragonal  basal  pinacoid,  97 
bipyramidal  class,  105,  153 
bipyramids,  first  order,  93 
second  order,  94 
third  order,  105 
bisphenoidal  class,  111,  153 
bisphenoids,  first  order,  102,  111 
second  order,  111 
third   order,   113 
hemihedrisms,  101 
prisms,  first  order,  96 
second  order,  96 
third  order,  105 
pyramidal  class,   110,   153 
pyramids,  first  order,  110 
second  order,  110 
third  order,  110 
scalenohedral  class,  102,  153 
scalenohedrons,    103 
system,  5,  92,  153 
trapezohedral  class,  108,  153 
trapezohedrons,  108 
trisoctahedron,    20 
tristetrahedron,   28 
Tetrahedral  group,  26 
hemihedrism,  26 

pentagonal  dodecahedral  class,  38, 150 
dodecahedron,  39 


Tetrahedrite,  31,  150 
Tetrahedron,   26 
Tetrahexahedron,  21 
Thirty-two  classes  of  crystals,  14,  150 
symmetry,  14,  150 

Thorium  sulphate,  131 
Threelings,  148 
Topaz,   8,   121 
Tourmaline,  74,  152 
Trapezohedral   group,   hexagonal,  68 
tetragonal,  108 
trigonal,  81 
hemihedrism,  hexagonal,   56,  68 

tetragonal,    101,    108 
tetartohedrism,  76 

Trapezohedron,  20 
Trapezohedrons,   hexagonal,   68 
tetragonal,  101,  108 

Triclinic  system,  5,  138,  156 
Trigonal  bipyramidal  class,  76,   152 
bipyramids,  first  order,  56,  77 

second  order,  77,  81 

third  order,  78 
hemihedrism,  55,  56 
prisms,  first  order,  56,  72,  77 

second  order,   77,   83 

third  order,  78 
pyramids,  first  order,  70,  90 

second  order,  90 

third  order,  90 
pyramidal  class,  89,  152 
rhombohedral  class,  85 
tetartohedrism,  76 
trapezohedral  class,  81,  152 
trapezohedrons,   81 
trisoctahedron,  20 
tristetrahedron,   28 

Trillings,  148,  149 
Trimetric  system,  115 
Trisoctahedron,  20 

tetragonal,  20 

trigonal,  20 

Tri-rhombohedral  group,  85 
Tristetrahedron,  tetragonal,  27 
trigonal,  28 

Twin  crystals,  144 


162 


INDEX. 


Twinning  axis,  145 
law,  145 
plane,  144 

Twins,  144 

contact,  145 
juxtaposition,  145 
penetration,  145 
reflection.  145 
rotation,  145 


Unit  form,  7 
Urea,  104 

Weiss,  Prof.  C.  S.,  11 

symbols,  11 
Wulfenite,  111,  153 

Zincite,  55,  151 
Zircon,  8,  98,  147 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 

This  book  is  DUE  on  the  last  date  stamped  below. 
Fij 


NOV  10  1947 

LIBRARY  USE 

JUL  2   1959 
REG'D  LD 


MAR161959B 


JUL 


JUN    81953LU 
JAN  171954  LU 

LD  21-100m-12,'46(A2012si6)4120 


16Ju!'5SHj 


JUL  16 


LIEHASY  'J 

AUG 


LD 


AUG 


REC'D 

AUG 


LD 


•V.    .. 

UNIVERSITY  OF  CALIFORNIA  LIBRARY 


